The Wandering Jew is more than an attractive plant and the practice of "wandering" was not unique to Ukraine.  It happened throughout the old country.  Contributing factors could include:

1.  traditionally large families...a numbers game, as a descendant was bound to leave the original parental shtetl eventually.
2.  economic necessity...many of our ancestors were poor/impoverished and they pursued a better living standard.
3.  marital considerations...this often resulted in clusters of the extended family appearing in surrounding shtetls, with movement back and forth over generations.  The larger the family relative to the size of the homestead shtetl, the more likely this was to happen.
4.  anti-Semitism...was there a safer place to live?
5.  education and training...if it was not sufficiently available in the homestead shtetl.  Skilled trade apprentice or rabbinical training are examples.
6.  wars...parts of the old country had frequent conflicts, which prompted civilian movement.
7.  death of the patriarch...a subset of economic necessity, where the widow and children might move.
8.  Russian May Laws...its hard to conscript a Jewish male for two decades of cannon fodder, if you cannot find him.  (This sometimes accompanied a surname change.)

Ken Domeshek.  Houston, TX.