This school plugged along in a small way for a couple hundred years. Then Emperor Ashoka around 300 BC converted to Buddhism. He worked to promote Buddhism across his empire, which extended through most of modern day India and beyond into Pakistan etc. He also sent Buddhist missionaries to distant places such as Greece. Buddhism continued to grow in India, with ups and downs, over the next centuries. By 700 AD there were several great Buddhist universities in India with over 10,000 students each. Buddhism spread to Sri Lanka, Burma, Cambodia, and Indonesia. Buddhism also spread to the northwest, into Afghanistan and up into Turkestan, and from there along the silk route into China. Buddhism reached China perhaps in 200AD. Buddhism got to Japan around 700AD, and Tibet about the same time.
Buddhism coexisted with other local religions in most places, e.g. Vedic religion in India, and Taoism and Confucianism in China. There tended to be a lot of borrowing back and forth across religions over the centuries. Buddhism even picked up elements of Greek culture from the remains of the Alexandrian empire around Afghanistan. For example, the typical statues of Buddhas are based on Greek sculptures of Apollo.
Eventually the spread of Islam from the West in the period 700-1300 AD wiped out Buddhism in India and the Northwest, but Buddhism continued to thrive in Ceylon, Burma, Cambodia, China, Japan, Tibet, and Mongolia. With this temporal and geographic scope, Buddhism is clearly one of the great world religions. As one might expect, Buddhism has accumulated a very wide spectrum of philosophical schools. I hope my quick sketch here will not be too far out on the fringe.
The core of the Buddha's teaching is that suffering arises from confusion, from the emotional turmoil caused by this confusion, and from the unskillful actions driven by that mix. The path taught by the Buddha attempts to liberate beings from suffering, addressing unskillful action with a code of conduct, emotional turmoil with meditation, and confusion with doctrine about the nature of things.
The doctrines about the nature of things are perhaps most relevant to the discussions here. Our general widespread confusion is one that takes objects to behave and exist in ways that they actually don't. We tend to perceive and conceptualize things as if they existed as stable, enduring, isolated, and well-defined. But in fact things are constantly changing and only exist as a parts of patterns, interrelated with other things.
The Buddha's teachings on the nature of things were memorized and interpreted and eventually written down in various ways, forming classically 18 different schools. Typically these analyse the objects of everyday life by reducing them to composite structures built up from elements, variously categorized by the different schools. Generally the categories fall coarsely into the five heaps of form, feeling, perception, conception, and consciousness. The elements in these categories are generally held to exist only momentarily, continuous existence being an illusion based on the successive momentary appearance of similar elementary components.
By 100 AD or so the Madhyamika school had emerged, championed by Nagarjuna. Nagarjuna argued that even these various elementary components do not exist as distinct isolated well-defined entities, but only exist by virtue of participating in a pattern of interrelationship.
I hope this very short explanation serves to reduce confusion and does not create confusion or emotional turmoil or lead to unskillful action!!!