I have a person in my own family who's aware of his past-birth. He knew it by age 7 only. When family started investigation 'they found it all true. How he has died in his last birth, how much of children he had, what work he was doing when he died and etc. I want to know what the buddha said about rebirth and transfer of memories to a new born? Moreover I wish to get more examples.
As for conciseness it arises and passes. At death it passes in one body and arise in another. This process don't end at death or with brain damage or death. As this instance passes away the next state arises with persistent sense impressions.
Sense impressions are tied to the mental body while conceptual impressions are tied to the physical body. So if you get brain damaged you cannot think and perhaps learn new concepts but past impressions stay, [Mahā,nidāna Sutta] these sense impressions do persist beyond death as part of the Bhavanga Citta. Recallability depends on the level of hindrances. [(Nīvarana) Sangarava Sutta] The mental impressions that remain are part of the mind (Nama) which does not die at death but passes away in one body and arise in another. For further understanding perhaps you can study the Buddhist rebirth process.
Jhana meditators can recall past lives since sense impressions have persisted. 5 Hindrances is one case of we cannot remember and as meditation overcomes this you can remember better. [(Nīvarana) Sangarava Sutta]
The Khajjaniya Sutta explains whatever experiences from the past are recollected, they were never "I", "me" or "mine". The Khajjaniya Sutta explains the mind was mistaken in the past, i.e., ignorant, to regard experiences as "I", "me" or "mine". You would probably need to look elsewhere for an explanation of reincarnation since the Khajjaniya Sutta refutes the idea of "I existed in the past".
Note: The Pali term "pubbe nivāsaṃ" at the start of the Khajjaniya Sutta does not mean "past lives". It means "previous dwellings". In his new translations, Bhikkhu Bodhi uses the term "past abodes".
The term similar term 'nivesā' ('fixations') is also found in the Haliddakani Sutta, which provides a good explanation of the idea of "dwellings".