Did your family pass through Antwerp, Belgium on their immigration journey?
If so, then you might find information in the Foreigner Index at the Felix Archive in Antwerp, which contains files of foreigners who lived there from 1840-1930. Many of these individuals were Eastern European Jews.
Previously researchers could find names, file numbers, date and place of birth at the archive, but now these details can be found at FamilySearch.org using a name search.
Here's a portion of a page, with Joseph Aaronowicz from Czernowitz highlighted:
Read more about the Police Immigrant Index here.
While I did not find any of my TALALAY in the index, I did find numerous FINK that might be related. Now all I need is more time to check out those names.
To search this index, click here.