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I think early Buddhist missionaries sent from India by Ashoka the Great did make attempts to transmit the religion to the west, and they even went as far west as Greece and Egypt in ancient times... but it wasn't too successful. I think its because the Far East was more willing to accept Buddhism, while South-East Asia was more willing to accept Indian culture in general.
If SE Asian cultures weren't Indianized, they would have been completely Sinicized, since that's the only other option there would be for SE Asians. Of course there's always the possibility of SE Asia being Islamicized, since Indonesia and Malaysia are already Muslim countries.
Buddhism was extremely successful in Central Asia, but I don't think it went as far as west. Gandhara is a perfect example. Greek Kings completely converted into buddhism/indian culture and even went to war after fall of Mauryan empire against Sungha Dynasty to protect buddhism.