How's this? Buddhism expands in India at the expense of Hinduism as IOTL, spreads to Tibet and East Asia, then the White Huns or somebody like them (Saka, Tocharians) conquer Persia and spread Buddhism there. Eventually, you have Buddhism dominating from the eastern Mediterranean to Japan. Is this possible or ASB?
How would Western variants of Buddhism diverge from the OTL eastern ones, in any case?
How would Western variants of Buddhism diverge from the OTL eastern ones, in any case?