What happened to the Buddha's teaching after him is a story of division, vast geographic spread, and eventual near-disappearance from the land of its origin.
The tradition records councils convened to fix the teaching, and within a few centuries the community had divided into schools. The surviving fault line runs between Theravada, the way of the elders, which preserves the Pali canon and predominates in Sri Lanka and Southeast Asia, and the Mahayana, the great vehicle, which developed new scriptures and the ideal of the bodhisattva and spread through China, Korea, Japan and Vietnam. A later Vajrayana or tantric stream became central in Tibet and Mongolia. Buddhism travelled the trade routes as far as it did partly because it carried no requirement of ethnic or territorial belonging.
In India itself it declined from roughly the end of the first millennium and had largely disappeared by about 1200, under a combination of pressures still debated by historians. Hinduism, meanwhile, absorbed the founder in a characteristic way: several Puranas list the Buddha as an avatar of Vishnu, commonly the ninth, but frame the descent polemically, as Vishnu taking form to lead the unworthy away from the Vedas, honouring the Buddha and neutralising him in one move.
The Indian story has a modern coda. In 1956 the jurist B. R. Ambedkar, principal architect of the Indian constitution and a leader of the Dalits, converted to Buddhism with hundreds of thousands of followers, founding a socially engaged Navayana movement and returning a Buddhist presence to the country the Buddha had lived in.