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The Zoroastrian emigration from Persia to Bombay is one.
See this article for more details: Parsi (Wikipedia).
Upvote:2
In addition to all the examples listed in the comments:
One of the most famous and studied example is the Visigoths, Ostrogoths (and other tribes) fleeing the Huns in the 4-5th centuries AD and entering the Roman empire. Any book covering fall of the (Western) Roman empire will discuss this in great detail.