How did buddhism spread across east asia?

 Being primarily a teacher, Buddha traveled to nearby kingdoms to share his insights with those who were receptive and interested. Likewise, he instructed his monks to go forth in the world and present his teachings. He did not ask others to criticize and give up their own religion and convert to a new one, for he was not seeking to establish his own religion. He was merely trying to help others overcome the unhappiness and suffering that they were creating for themselves because of their lack of understanding. 

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