The so-called Mahayana Buddhism spread out to Central Asia first from India as far as I know. Principal inhabitants of Central Asia at the time were Caucasians. From there, it spread to Steppe nomads in the East. All barbarian dynasties in N. China after 304 AD adopted Buddhism this way. I agree that Tocharian scripts could be used for Buddhist literatures among these societies. These Northern dynasties sent missionaries to further East and South. That is how all ancient Korean kingdoms got their Buddhism, even though there is no known reference to Tocharian scripts in Korea up to now.
In China, Buddhism started to overshadow Confucianism as a state ideology. China was on the verge of civilizational collapse. Chinese population plummeted from the peak of 100 million in the late Han dynasty to mere 8 million by the time the barbarian dynasties establish themselves, according to a recent research on the historical demographic change in China. Expanding northern dynasties also brought in new cultural elements from the West and the Central Asia. China was on the defensive on both demographic and cultural front. It took China 300 years to reconstitute herself, absorbing truly diverse ethnic groups and culture from all directions, leading to the rise of Tang dynasty which was quite international in its character. It is also principally a Buddhist dynasty.