Whether doing Kiruv, or running a yeshiva aimed at Baalei Tseuvah, one has to be prepared for the possibility that someone raised in a non-frum background is not Jewish, or at least, there is no way of determining if they are Jewish. One should note that in the case of Baalei Tseuvah who are already doing mitsvos, and then discovers they may, or perhaps definitely, are not Jews, conversion is greatly simplified by the fact they are already adopting a Jewish lifestyle. At this point, anyone who is of German descent (i.e. from the 1848 waive of immigration) or Sefardi (from those fleeing the Spanish and Portuguese in the 16th and 17th centuries), whose family has not remained at least nominally Orthodox, probably is a safek goy, and this will soon be the case, if it isn’t already, of the waive of East European Ashkenazim who came in the late 19th and early 20th century.
On the bright side, during the first generation or two of going “off the derekh” people often had arguably valid marriages but rarely bother with a “get” when they got divorced – but if it turns out they are goyim that solves the problem (since it is lot easier to convert that to deal with mamzerus).
There is a view that it is prohibited from trying to convince goyim to convert, but it isn’t clear how that appears to non-Jews of Jewish descent. It needs to be remembered that until recently, it was a capital offense for a goy to become a ger, so any Jews openly encouraging goyim to convert would be executed.