As buddhism coexsits with shinto in Japan pretty much the same happened before in the continental Asia. Buddhism at the beggining was totally denying to be a religion rather than a way of life, and was also extensively tolerant with local religious practices in the countries where missionary monks arrived. Hinduism as nowadays was innitially conceived as a reaction to the Buddhism and especially Mahabharata and the Krishna cult were aimed to stop the popularity of the new current which was perceived by the Brahman cast as a threat to the accepted order (as Buddhism plainly rejected not the Hindu religion in itself but the rigid stratification in social layers). In time India came back to the Hindu religion, Buddhism being now a pretty small minority, but the other countries converted to Buddhism and especially those in continental Asia suffered a massive infusion of Hinduism through the third process - of the Buddhist monks trying to peacefully assymilate back into the Buddhism the Hinduist reaction to their cult rather than opposing to it by aggressive means. Things were not so smooth as described, religious war was raging constantly here and there but nevertheless Buddhism proved as a solid ground to integrate many currents which were innitially opposed. Ramayana and Mahabharata as we know them today are versions dating from as late as maybe 16-th century A.C. whilst the Hinduist reaction to Buddhism was at climax around 7-800 A.C. so it is rather hard to know how things really happened, the more that early Buddhism had a point of honour in not writing any dogmatic writings but conserve a pure master-to-disciple path of transmitting tradition.