Anyhow, the BNS specifically mentions rules for travel #37, I haven't figured out if this means one is supposed to go on a pilgrimage.
I agree with Matthew's answer.
Note that the reference for #37 says (unlike most of the other precepts),
- Traveling in Dangerous Areas
[As a cleric], a disciple of the Buddha should engage in
So I think this is explicitly for clerics, who have "gone forth", i.e. are homeless and are maybe (instead of living in a fixed place/home) travelling from place to place, perhaps begging and so on, i.e. where travelling is their usual way of life.
Those customs (annual "ascetic practices") are not only Mahayana/Chinese -- Wikipedia's Vassa article says,
The Vassa tradition predates the time of Gautama Buddha. It was a long-standing custom for mendicant ascetics (see Sramana) in India not to travel during the rainy season as they may unintentionally harm crops, insects or even themselves during their travels. Many Buddhist ascetics live in regions which lack a rainy season. Consequently, there are places where Vassa may not be typically observed.
The Śramaṇa article mentions,
It was as a śramaṇa that the Buddha left his father's palace and practised austerities. ...
The Buddhist Sramanic movement chose moderate ascetic lifestyle. This was in contrast to ...
The śramaṇa idea of wandering began to change early in Buddhism. The renunciates started living in vihāras, at first during vassa, the rainy season, but eventually permanently. In medieval Jainism the tradition of wandering also waned, but it was revived in the 19th century. Similar changes have regularly occurred in Buddhism.
Here's more about it from The Bhikkhus' Rules -- A Guide for Laypeople,
The Rains Retreat
The bhikkhu's year is structured around the three months from July to October. In Asia this is the time of the monsoon season — the central period of the agricultural year — when the paddy fields are flooded and the main rice crop is planted. In the Buddha's time (and until modern times), people were less likely to travel around during this period because the roads were bad and there was a danger of crop damage. So the bhikkhus likewise suspended their mendicant wanderings and had to settle in one place.
A bhikkhu must make a formal determination to be resident at dawn every day in that place for the whole three month period. (There are exceptional circumstances when he may be allowed to be away, but even then he should return within seven days.) These three months are often a special time of study or meditation and so are sometimes known as the Rains Retreat or Rains Residence. This is also the normal time when the young men of South East Asia become monks for the traditional three month period (see above).
A bhikkhu often measures the length of time he has been a monk according to how many Rains Residences he has undertaken. Therefore instead of saying he has been 'ordained seven years' he might say he has been ordained for 'seven Rains.'
Like the referenced 37th precept, the Buddhist Monastic Code (Patimokkha Rules) also contains various references to avoiding danger, including,
the origin story to this rule indicates that this period was a dangerous time for
bhikkhus living in wilderness areas, as thieves were active—perhaps because
they knew that bhikkhus had just received new requisites, or simply because
now that roads had become passable it was time to get back to their work.