We well know that the great religions were born on the yellow continent. Judaism, Christianity and Islam are proof of this. And the buddhism does not escape from it. This religion never developed as a missionary movement. The teachings of buddhism gradually spread from regions such as India and Tibet to other parts of the continent, and then to the entire world.
Each place where Buddhism was taught, the doctrine was transformed according to the traditions of the place, thus creating a series of currents.
If we talk about Buddhism in Asia It is worth mentioning that this religion spread peacefully and in different ways, and it tells the story that it was Shakyamuni Buddha, holy man from Nepal who set the precedent. This sacred being and spiritual guide traveled to the closest realms to share his profound intentions and that is how he indoctrinated monks with the purpose that they would go around the world to expose his teachings.
At present, different theories about the expansion of Buddhism in Asian lands, one of the most acceptable theories being the one that mentions the ambulatory merchants and traders of the famous Silk Routes, like the main expansionists of Buddhist beliefs, especially from the Theravada line. This is how we can find sources of Buddhism even in countries Middle East that, as you well know, most profess the Islamic religion.
As for the Mahayana stream (vehicle to achieve nirvana), most of the faithful are found in the areas of China, Vietnam, Korea, Japan, India, Indonesia, Nepal, Tibet, Mongolia, and even the area from Siberia.