Good question! As you said, Eastern Ashkenazim are probably mostly descended from Central European Ashkenazim who moved east in the Middle Ages. Central European Ashkenazim probably got there from the Mediterranean. It's hard to know for certain, since early Ashkenazi communities have very scant documentation besides their existence. But it's the most likely geographic course of events, and loanwords and names in Yiddish that have Latin origins give some evidence to it. Probably during the Late Roman Empire, Jews kind of diffused from Italy into modern-day France and Germany.

As for how Jews got there, the default assumption is usually that they were refugees or prisoners following the Great Jewish Revolt in the late 1st century CE. It's often assumed from the terminology of "exile" that most or all Jews were forcibly removed from Judea and enslaved in Rome, but we know that there existed a Jewish population in Rome (and in many other places) before the revolt, and a significant community persisted for centuries afterwards (enough to revolt again within a century, and there was still a large community after that). It's also hard to imagine Judaism persisting in recognizable form entirely among enslaved people for centuries, though I suppose it's possible. The most likely thing is that the Jewish population in Italy, which existed for a century or two before the great revolt, had some addition to its numbers from people forcibly brought from Judea by the Romans. But a lot of people probably just wanted to leave a war-torn area that was economically not doing very well for the next several centuries, and ended up in Italy, and eventually moved north to become Ashkenazim.

It's worth remembering that it's not just the Ashkenazi population that was founded this way. The Jewish populations in Spain, Greece, and North Africa (and Italy itself, for that matter) would've been founded the same way. But overall, there's a lot of uncertainty about that this looked like--written evidence for early communities is generally simply an attestation of a community's existence, finding out how they got there is pretty rare, so we're left to fill in a lot of blanks.

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Sources, and further information: The Cambridge Medieval History vol 1 has a nice chapter on Early Medieval Jews. Unfortunately because of the lack of sources it can't say how exactly Jews moved around. It does suggest that the migrations may've been driven by trade, as a large share of Jews were merchants, particularly in Northern Europe. It does helpfully lay out when Jewish populations are attested where in Europe.

Bernard Bachrach's "Early Medieval Jewish Policy in Western Europe" has information on how European rulers dealt with Jewish populations in their territories, with information about what this says about social relationships between Jews and non-Jews.

Erich Grun's "Diaspora: Jews Amidst Greeks and Romans" and "The Jews Under Roman Rule: From Pompey to Diocletian : a Study in Political Relations" by E Mary Smallwood both discuss how Jewish populations outside Israel began in areas ruled by Rome.

And finally, "Between East and West: A History of the Jews of North Africa" by André Chouraqui is a bit orthogonal to your question but may be of interest. In its discussion of the beginnings of the Jewish community in North Africa it has some discussion of evidence for Jews moving around and establishing communities in the Mediterranean during the late antiquity.