While /u/longjumping-mobile-9 talks about the "Were they migrants or local coverts" part of your question, you may be interested in the answers from myself and /u/idjet for the other parts of your question:
As /u/longjumping-mobile-9 mentions, there's quite a debate and there's very thin evidence for surprisingly large stretches of time in surprisingly broad swathes of territory. The story of the Jews traveling to Eastern Europe necessarily starts with the Jews moving into France and Germany in post-Roman times. The gradually expansion into Eastern Europe from both the South (tied to Byzatinum) and the East makes a lot of sense in that context—even if I have never gotten a satisfying explanation for the demographic lopsidedness or, even, for sure how they all ended up speaking Yiddish.
(If you don't have a great sense of European geography—if you don't know the Rhone from the the Rhine—it may help to follow along with a map)