In an effort to understand the concept of lovingkindness it would help me to see how the Buddha explicitly described lovingkindness and specific acts of his unconditional compassion. I've heard from secondary sources that he'd throw out his "net" of compassion to walk 200 miles to teach someone, but I'm looking for references to primary or well known secondary (e.g. Bhikku Bodhi, Analayo) sources, please.
Shakyamuni Tathagata Lord Buddha spent his whole life expressing loving-kindness and compassion to all living beings. You can understand this by reading about Shakyamuni Tathagata Lord Buddha's daily routine during his remaining 45 years of life after the enlightenment. There are many stories and incidents which highlight these spiritual qualities of Lord Buddha. Here I've mentioned few of them with references;
Story of Patacara
Born the daughter of a wealthy banker, Patacara fell in love with one of her father's servants and ran off to live happily with him in a forest hamlet. Then, through a series of tragic accidents, she lost first her husband, then two sons, and finally her parents and brother.
Wandering destitute, naked and mad with grief, she in time met the Buddha face to face, who showed her kindness when others would scorn her. She heard his teaching, joined the order of nuns, and went on to become one of the arahants, or worthy ones. Patacara helped many other women who were overcome by grief regain their sanity and quench the pain of their loss.
This tender poem of loss and recovery was (probably) composed by Patacara;
"My son!" you weep, for one whose path
You do not really comprehend
—Whether he's coming or going.
From where has that son of yours come?
And yet for one whose path you know...
For him you do not grieve at all
— Whether he's coming or going.
Such is the nature of creatures.
Uninvited, he came from there;
Unpermitted, he's gone from here.
And having come from who knows where,
He lived for but a few short days.
But though gone from here by one (path),
He goes from there by another.
Departed, with a human form,
He will go flowing on and on.
As he has come, so has he gone.
What is there here to grieve about?
My thorn, indeed, has been removed!
Buried in the heart, so hard to see.
That grief which had overcome me
— Grief for my son — has been dispelled.
Today the thorn has been removed.
Without hunger, I've become quenched.
To Buddha, Dhamma and Sangha,
I go to the sage for refuge.
For more detailed story of Patacara, please read: The Story of Lady Patacara
Story of Sunita
In Savatthi there was a scavenger named Sunita. He was a road-sweeper and barely earned enough to feed himself. Sunita slept on the roadside, for he did not have a house to go to. He saw other people enjoying themselves but he could not mix with them because these people called him an outcast. Whenever a higher caste person went on the road Sunita had to run and hide so his shadow did not fall on them. If he was not quick enough he would be scolded and beaten. Poor Sunita lived a miserable life.
One day, as he was sweeping a dirty, dusty road, Sunita saw the Buddha with thousands of followers coming towards him. His heart was filled with joy and fear and finding no place to hide he just stood, joining his palms in respect. The Buddha stopped and spoke to poor Sunita in a sweet, gentle voice saying, "My dear friend, would you like to leave this work and follow me?"
Nobody had ever spoken to Sunita like this before. His heart was filled with joy and his eyes with tears. "O, most venerable Sir, I have always received orders but never a kind word. If you accept a dirty and miserable scavenger like me I will follow you."
So the Buddha ordained Sunita and took him along with the other monks. From that day forth no one knew what Sunita's caste was, and nobody treated him with disgust and cruelty. Everybody, even kings, ministers and commanders, respected him.
Reference: Sunita, the scavenger
Here's a poem composed by Sunita after he became one of the Arahants; [from the book Theragatha, Khuddaka Nikaya, Tipitaka]
In a lowly family I was born,
poor, with next to no food.
My work was degrading:
I gathered the spoiled,
the withered flowers from shrines
and threw them away.
People found me disgusting,
despised me, disparaged me.
Lowering my heart,
I showed reverence to many.
Then I saw the One Self-awakened,
arrayed with a squadron of monks,
the Great Hero, entering the city,
supreme, of the Magadhans.
Throwing down my carrying pole,
I approached him to do reverence.
He — the supreme man — stood still
out of sympathy just for me.
After paying homage
to the feet of the teacher,
I stood to one side
& requested the Going Forth from him,
supreme among all living beings.
The compassionate Teacher,
sympathetic to all the world, said:
That was my formal Acceptance.
Alone, I stayed in the wilds,
I followed the Teacher's words,
just as he, the Conqueror, had taught me.
In the first watch of the night,
I recollected previous lives;
in the middle watch,
purified the divine eye;
in the last,
burst the mass of darkness.
Then, as night was ending
& the sun returning,
Indra & Brahma came to pay homage to me,
hands palm-to-palm at their hearts:
"Homage to you, O thoroughbred of men,
Homage to you, O man supreme,
whose fermentations are ended.
You, dear sir, are worthy of offerings."
Seeing me, arrayed with a squadron of devas,
the Teacher smiled & said:
"Through austerity, celibacy,
restraint, & self-control:
That's how one is a brahman.
He is a brahman supreme."
Refernce: Sunita, the outcaste (translated by Thanissaro Bhikkhu)
Story of Putigatta-Tissa thera (Tissa, the thera with a stinking body)
On one occasion the Buddha was delivering a sermon to an assembly of bhikkhus/monks, bhikkhunis (nuns), upasakas and upasikas. That day a householder of Savatthi was listening to sermon. He realized that the household life was full of difficulties and that the life of a monk is peaceful. He subsequently entered the Sangha.
While training himself in the spheres of morality, concentration and wisdom, the bhikkhu became ill. There appeared on his body boils that became bigger and bigger until they burst and turned into ugly ulcers. When these sores burst, his upper and lower robes became sticky and stained with pus and blood, and his whole body was stinking. For this reason, he was known as Putigattatissa, Tissa the thera with stinking body.
His bones decayed and gave way. His body was rotting and even his relatives and disciples ceased to look after him.
As the Buddha surveyed the universe with the his Dibbacakkhu (divine eyes), the thera appeared in his vision. He saw the sorrowful state of the thera, who had been abandoned by his resident pupils on account of his stinking body. At the same time, he also knew that Tissa would soon attain arahantship.
So, the Buddha proceeded to the fire-shed, close to the place where the thera was staying. There, he boiled some water, and then going, to where the thera was lying down, took hold of the edge of the couch. It was then only that the resident pupils gathered round the thera, and as instructed by the Buddha, they carried the thera to the fire-shed, where he was washed and bathed. While he was being bathed, his upper and lower robes were washed and dried.
After the bath, the thera became fresh in body and mind and soon developed one-pointedness of concentration. Standing at the head of the couch, the Buddha said to him that this body when devoid of life would be as useless as a log and would be laid on the earth.
Then the Buddha spoke in verse as follows:
“aciraṃ vat’ ayaṃ kāyo paṭhaviṃ adhisessati
chuddho apetaviññāṇo niratthaṃ va kaliṅgaraṃ”
Before long, alas, this body,
deprived of consciousness,
will lie on the earth,
discarded like a useless log.
At the end of the discourse Thera Tissa attained arahatship together with Analytical Insight, and soon passed away. The Buddha attended to the obsequies of the thera, and got a stupa erected to enshrine his ashes.
When the monks questioned the Buddha as to the next birth of the thera. He answered that he would no more be born, and that he passed away after becoming an arahat.
Story of Rajjumala
Rajjumala was a slave in the village of Gayá. Her mistress disliked her and ill treated her in every way. One day, in order to escape being pulled by her hair, she had her head shaved; but her mistress then had a rope tied round her head, with which she pulled her about; hence her name.
Unable to bear her life any longer, the slave went into the village near by, intending to commit suicide, but there she found the Buddha waiting for her, and he preached to her. Rajjumálá, became a sotápanna, and then returned to her mistress, who, having heard her story, visited the Buddha and became his follower.
The Buddha explained that the two women had had their positions reversed in a previous birth, and that the then slave, who was the mistress in the present birth, had vowed vengeance for the cruelty inflicted upon her.
Rajjumálá was freed and was born after death in Távatimsa.
For more details, please read: Rajjumala’s Mansion, Khuddaka Nikaya 4.12, Tipitaka
Story of Sopaka
Note: There are two versions of the story of sopaka mentioned in different sources. So please research further and find the one which is correct. Here, I've mentioned both.
Sopaka was the son of a very poor woman of Sāvatthi. While in labour his mother fell into a long and deep swoon, and her kinsfolk, thinking her dead, took her to the cemetery and prepared for cremation. But a spirit prevented the fire from burning with a storm of wind and rain, and they went away. The child was safely born and the mother died. The spirit, in human shape, took the child and put it in the watchman's hut, feeding it for a time. After that the watchman adopted it, and the child grew up with the watchman's son, Suppiya (q.v.). He was called Sopāka, (the "waif") because he was born in the cemetery. When he was seven years old he came under the notice of the Buddha, who visited him in the cemetery. Gladdened by the Buddha's teaching, he sought his father's consent and entered the Order. The Buddha gave him, as his subject of meditation, the thought of mettā, and Sopika, developing insight, soon attained arahantship.
Sopaka was born as the child of a cemetery keeper and was therefore called Sopāka. Others say that he was born in a trader's family and that Sopāka was merely a name. Four months after birth his father died suddenly and he was adopted by his uncle. When he was only seven years old, his uncle took him to a charnel field because he quarrelled with his cousin, bound his hands, and tied him fast to a corpse, hoping that the jackals would eat him. At midnight the jackals came and the child started crying. The Buddha, seeing Sopāka's destiny for arahantship, sent a ray of glory, and, by the Buddha's power, the boy broke his bonds and stood before the Buddha's Gandhakuti, a sotāpanna. His mother started seeking for him, and the uncle telling her nothing, she came to the Buddha, thinking "The Buddhas know all, past, present and future." When she came, the Buddha, by his iddhi-power, made the boy invisible and taught her the Dhamma, saying that sons are no shelter, blood bonds no refuge. As she listened she became a sotāpanna and the boy an arahant. Then the Buddha revealed the boy's presence to his mother, and she allowed him to enter the Order. Some time later the Buddha, wishing to confer on him the higher ordination, asked him the questions which came to be known as the "Kumārapañhā" Sopāka answered these, and the Buddha, satisfied, gave him the upasampadā.
Reference: Sopāka Thera
For further reading: The Story of Poor Sopaka
Story of Mattakundali
Maṭṭakuṇdali was a young Brāhmin, whose father, Adinnapubbaka, was very miserly and never gave anything in charity. Even the gold ornaments for his only son were made by himself to save payment for workmanship. When his son fell ill, no physician was consulted, until it was too late. When he realized that his son was dying, he had the youth carried outside on to the verandah, so that people coming to his house would not see his possessions.
On that morning, the Buddha arising early from his deep meditation of compassion saw, in his Net of Knowledge, Maṭṭakuṇdali lying on the verandah. So when entering Sāvatthi for alms-food with his disciples, the Buddha stood near the door of the Brāhmin Adinnapubbaka. The Buddha sent forth a ray of light to attract the attention of the youth, who was facing the interior of the house. The youth saw the Buddha; and as he was very weak he could only profess his faith mentally. But that was enough. When he passed away with his heart in devotion to the Buddha he was reborn in the Tāvatiṃsa celestial world.
From his celestial abode the young Maṭṭakuṇdali, seeing his father mourning over him at the cemetery, appeared to the old man in the likeness of his old self. He told his father about his rebirth in the Tāvatiṃsa world and also urged him to approach and invite the Buddha to a meal. At the house of Adinnapubbaka the question of whether one could or could not be reborn in a celestial world simply by mentally professing profound faith in the Buddha, without giving in charity or observing the moral precepts, was brought up. So the Buddha invited Maṭṭakuṇdali to appear in person; Maṭṭakuṇdali then appeared in his celestial ornaments and told them about his rebirth in the Tāvatiṃsa realm. Only then, the listeners became convinced that the son of the Brāhmin Adinnapubbaka, by simply devoting his mind to the Buddha, had attained much glory.
Reference: The Story of Maṭṭakuṇdali
You've asked about metta.
The deepest example of metta does not use the word metta.
In SN6.1, we hear that Brahmā Sahampati kneels and asks the Buddha to teach the Dhamma.
Sir, let the Blessed One teach the Dhamma! Let the Holy One teach the Dhamma!
To which the Buddha humbly replied:
Thinking it would be troublesome, Brahmā, I did not teach the sophisticated, sublime Dhamma among humans.
The Buddha spent the rest of his life teaching. Now that is the very definition of metta.