Well kind of. There was plenty of movement, yes.
But the borders and empire structure also moved while people stayed out.
My grandfather was effectively Polish, fought for the Germans in WWI, and then made it out by the skin of his teeth never having left the same billiard except to see service in France.
Many Jews served in the Great War. It didn’t help them much or at all later on.
The irony is that when Poland was divided between Germany, Austria-Hungary and Russia, the best area for the Jews to be in was the German part.