The cultures of China, Japan, and Korea were in close contact for millennia before they had any real interaction with the West. That's plenty of time for significant cultural exchange, even if the process is extremely slow. In fact, the conventional wisdom is that East Asian countries (especially Korea and Japan) have Westernized shockingly quickly given that the bulk of that process has happened over just the last century.

Religions in particular can spread very quickly before then sticking very tightly. Buddhism is primarily practiced in East Asia but started in South Asia. Islam started in Arabia but has adherents from Nigeria to Indonesia. Christianity started in Jerusalem (kind of) and is now practiced by at least a substantial minority on every continent. By comparison, taking Confucianism across some mountains or a sea is pretty easy.