The fact that the Greek residence permit has now expired does not change anything about your asylum procedure in Germany. I guess that the Federal Office for Migration and Refugees rejected your family's application for asylum and ordered deportation to Greece or Sweden. If your family has not only filed a complaint against this decision, but has also filed an application for suspensive effect, then it depends on the decision of the administrative court on this application whether or not your family may remain in Germany.
If your family didn´t file such an application for suspensive effect or the administrative court rejects the application for suspensive effect, the immigration authority has 6 months to deport your Family to Greece (or Sweden). If they fail to do this within that time limit, your family can stay in Germany to conduct your asylum follow-up procedure.