A regular Friday night guest who has recently become frum would like to bring his (non-frum) parents to show them a Friday night dinner. However, it will mean they will drive on Shabbos to get to us (and to go home). Can we invite them?
Two sources that are stringent concerning this issue are Rav Moshe Feinstein (Iggros Moshe, Orach Chaim 1:99) and Rav Elyashiv zt”l (cited in Chashukei Chemmed Pesachim p. 189). These authorities rule that where an invitation for Shabbos will cause the invitee to violate Shabbos by driving over, it is not permitted to do so.
According to this approach, it will only be permitted to invite a non-observant Jew if there is no clear knowledge that he will be driving, even where the possibility (or even likelihood) of him doing so exists.
However, there are also more lenient opinions, as mentioned here (see in particular in the name of Rav Shlomo Zalman and Rav Zalman Nechemya). In particular, there is some room for leniency where the invitation is deemed important for purposes of kiruv.
Thus, it is hard to give a blanket ruling for all circumstances, and in practice each case must be evaluated on its own merits. In the case of the question, it is possible that the merits of the case are not sufficient, and that the purpose of being nice to the family etc. can be achieved with a weekday invitation, perhaps for Purim or for Chol Ha-Mo’ed.