One of the interesting things about the spread of Buddhism is that it took a very roundabout route through the northwest of the Indian subcontinent and Central Asia (what is now Afghanistan and Pakistan) into East Asia along the Silk Road routes -- so the form it reached China in (Mahayana) was a bit separated from the credo of the original Buddha.

The 'golden age' of Buddhism in the Indian subcontinent coincided with its institutionalization by the Mauryan Empire, especially during the reign of Asoka. In fact, he issued a series of multilingual (some are in Greek and Aramaic) edicts throughout the empire advertising the non-violent morals of Buddhism, that he had converted various peoples in his empire (including Greeks), and that he had sent missionaries as far away as Sri Lanka and the Hellenistic kingdoms of the west. Stupas and Buddhist seminaries were sponsored en masse -- though these were institutions that ultimately relied on imperial or communal support and funding. While Buddhism became entrenched in a few places, such as Gandhara (a region that now straddles Afghanistan and Pakistan, known for Graeco-Buddhist art), it largely petered out outside of a select few pilgrimage points, universities and monasteries. It just wasn't endearing enough to the common imagination, nor did it attract much imperial support in the bulk of the subcontinent. So Buddhism went on a steady decline in the subcontinent, coinciding with the emergence of Hinduism and Islam.

Back to Gandhara and the northwest frontier, however: Buddhism flourished thanks to the capture of the region by Graeco-Bactrians and Kushans, who again institutionalized the religion through stupa-construction and funding of Buddhist monasteries and scholars. Mahayana Buddhism also emerged here, especially through humanistic depictions of the Buddha (whereas Buddhism was originally aniconic) and the concept of boddhisattvas. One of the principal vajrapanis (or protectors of the Buddha) in Graeco-Buddhist art is Hercules, for example, while Gandharan Buddhas aesthetically resemble depictions of the Hellenistic god Apollo. This is also evident from the travel accounts of Chinese traveller Fa-hsien, who travelled to the region during Hephthalite rule (better known as the Huns). He mentions the Buddhist mythology endemic to the region (such as fables that the Buddha had slain dragons there) and stupas built by the Kushan emperor Kanishka I, whose reign also saw the flourishing of Graeco-Buddhist art and Silk Road trade with Han and Tang-dynasty China.

Also during this period was the migration of many Gandharans and the Gandharan language into what is now Xinjiang in western China. The Gandharan language was a Sanskritic one written in either Greek or Aramaic script at various points in history, but in China, it was principally the language of Buddhism. Through the many Gandhari-language scrolls peppered throughout Silk Road oases and Buddhist monasteries, it is evident that Gandharan Buddhism -- with its heavy Hellenistic and Iranian influence -- was the medium by which Buddhism spread to China. There were even some prolific figures like the Kashmiri monk Kumarajiva who translated works like the Diamond Sutra directly from Sanskrit to Chinese, introducing tens of thousands of Sanskritic loanwords into the Chinese language. Important also were an Iranian people called the Sogdians, who left Buddhist inscriptions all over the Hindu Kush and the Hunza Valley of northern Pakistan -- they were primarily Zoroastrian and eventually Muslim, but the artwork Sogdians constructed in China and along the Silk Road often blends Iranian, Chinese, Hellenistic and Buddhistic motifs and elements.

Finally, it's important to note that the period of greatest incursion of Buddhism was during the Tang-dynasty period; that dynasty is known widely for its cosmopolitanism, and was deeply engaged with the geo-political affairs of Central Asia, Iran and the Tibetan plateau. Their capital at Chang'an (modern Xi'an), for example, hosted great shrines, temples and monasteries of various 'Western' religions, such as Nestorian Christianity, Buddhism, Zoroastrianism and Islam, and a very multi-ethnic population that still characterizes western China today. Buddhist monasteries were sponsored by the state, and members of the royal family even converted to Buddhism during this period. The famous Buddhist monk Xuanzang even travelled throughout Central Asia and India to collect Sanskrit texts during this period, and was accordingly celebrated by the contemporary Tang emperor.

That cosmopolitanism ended with the devastating An Lushan Rebellion, which led to increasing xenophobia and the persecution of 'non-native' religions like Zoroastrianism, Christianity, Manichaeism and Islam. Although Buddhism did not escape repression, it was endearing enough to establish long-standing monasteries and seminaries and thus survived until the decentralization of the Tang dynasty, by which point it flourished with native Chinese artistic and mythological elements. What was different about Buddhism, as compared to Zoroastrianism, Nestorian Christianity, Islam and Manichaeism, was that it had very little backing among the Central Asian foes of the Tang dynasty; Manichaeism, for example, was sponsored by a Uighur khanate that had opposed the Tang, and Zoroastrianism was the principal religion of the Sogdians, who constituted a considerable antagonistic faction during the An Lushan rebellion. Christianity and Islam, likewise, was endearing to various Turkic tribes.

Sources:

The Silk Road: A New History by Valerie Hansen