(Adding to and borrowing from a previous answer. This is heavily focused on Western monasticism and Augustine, acknowledging the desert of the East as the origin of Christian asceticism)

It is true that the standard of asceticism came from the Egyptian deserts. But the idea of a monastic community rather than individuals banded together in an area or hermits, came initially, in large part, out of the West. There were exception to this, notably Pachomius (born in Egypt in the late 3rd century), who founded the first communal monastery. The hermitic and ascetic type of monasticism is called "eremitical" and was practiced more heavily in the East in Late Antiquity and the Early Middle Ages, whereas an order living together that stressed community is "cenobitic" and was mostly in the West at that time. There were some individuals like Basil and Ambrose who formed monastic communities, but the systemisation began largely with the "written rule" of Augustine in the late 4th and early 5th century.

Augustine had encountered Eastern, ascetic practices early on in his conversion in Athanasius' Life of Antony. While Augustine was moved by this account of fierce individualism, he more longed for and envisioned a group of people of the same sex living in a community dedicated to philosophical contemplation and the Christian worship of God. Much of this idea came from a time in his life when he lived in a community of like-minded friends at Cassiacum, where they all hung-out and discussed books and ideas and worshipped without too much focus on material possessions or worldly ambitions.

Augustine abandoned pursuits of political appointments in Italy and relocated to his home region of Northern Africa intending to pursue a life of philosophic contemplation like at his retreat at Cassiacum. But circumstances (revealed in the autobiographical Confessions) both personal and political meant that Augustine was made Bishop of Hippo. In this role he still pursued his goal of forming a monastic community.

As Augustine became more popular as a bishop, the population of his monastic community grew. This called for him to write some pretty pragmatic "rules", in addition to his theological teaching and thought, on how people should get on with one another in this sort of environment and what was expected of them in terms of renunciation and commitments. These principles of service, humility, and contemplation all formed the foundation of his specific "rules" and the core tenants of future monastic writings. He saw and addressed the potential for pride in people who were ascetics and did extreme acts of devotion and penance and who might falsely look on themselves as holier than fellow Christian according to their works. He believed that grace was at the centre of salvation and addressed pride both his sermons, theological writings, and again, in his "rule" for monastic living.

Augustine founded one of the first monastic communities for women and put his sister in charge of it.

This sort of community would be adopted, replicated, and suited to the specific environment's needs by others like Caesarius of Arles in 6th century Merovingian Gaul and other areas that saw the bishopric become more and more a stabilising political and communal force for their societies. Caesarius would write his own "order" that borrowed from Augustine's thought and from John Cassian's precepts for monastic life, and adding to them his own innovations. This would be another source of monastic, communal governance.

Also in the 6th century, Benedict of Nursia would write an important guide for monastic orders. Another 6th century momentous, monastic innovation was a strain of "Irish" monasticism, with Columbanus at the helm.

The writings, traditions, and physical communities of these early founders would be studied and employed throughout the Middle Ages, eventually becoming foundations of learning and community from which would arise the great centres and universities of learning like Oxford, Genoa, Paris, and others.

For further info check out:

The Emergence of Monasticism by Marilyn Dunn

R.A Markus: The End of Ancient Christianity

Or any of Peter Brown's more general books on Late Antique Christianity.

For more on Augustine's influence on Monasticism, see:

Augustine's Ideal of the Religious Life by Adolar Zumkeller. It is available translated from German into English.

Link to Augustine's Rule

Can I just chime in here?

I just got to thinking today - Hinduism and Buddhism have asceticism and monasticism, but they both believe in reincarnation generally, so a non-monk non-ascetic is motivated to support because someday you might get reincarnated and need to achieve salvation as a monk.

​While this is... um, logically consistent, it's not typically what most Buddhist and Hindu monasticism operate on, i.e. the idea that one might be reincarnated as a monk/yogin/sannyasin so to give to one is to potentially give to yourself in a future life. Buddhist culture of giving typically operates on the concept of "merit" which is based around (crudely speaking) scoring points towards the afterlife.

Merit is different from Karma. Karma has infiltrated our culture as a sort of spiritual plus and minus system usually revolving around unrelated events. "Karma" literally translates to "Action" or "Deed" and the spiritual concept can be epanded to "the law of cause and effect." Karma is more like how Action A leads to Effect B, not how Action A triggers a universal retribution/reward mechanism which leads to Result B.

Karma is distinguished from merit in that Karma can be positive, negative, neutral, or mixed. Merit, however, only comes in a single flavor: positive. While pretty much anything can be considered Karma (i.e. saving a bug from drowning in a pool, cutting someone off in traffic, committing your life to working in hospice care) there's really only a select few activities that are considered merit, usually things that are done with an express religious purpose, most notably giving to the Sangha or participating in Sangha activities (i.e. circumambulation, commissioning writings or printings).

Reincarnation in these settings is usually taken for granted. And while there are many more former and eventual ascetics as there are current ascetics, the idea that "it might happen in a future life" is usually not on people's minds so much as current meritorious practice.

Sources:

Richardson, Snellgrove, A Cultural History of Tibet

Stein, Tibetan Civilization