Is there a religious basis for the custom prevalent among the people of two villages—characterized by strong ties and kinship—of immediately departing after attending a funeral or offering condolences, and refusing to stay for hospitality with their relatives (who are not the deceased's family), on the grounds that it is inappropriate or impermissible on such an occasion?

The customs mentioned in the question have no evidence in Islamic law. They appear to be social and psychological customs and should not be considered sacred religious law, especially since they involve a shortcomings in maintaining kinship ties. Customs that contradict Islamic law or its objectives, or lead to shortcomings, must be abandoned and changed wisely. The basic rule for customs is permissibility unless prohibited by Islamic law. Customs must be presented to Islamic law: what it approves is permissible, and what it does not is not. Adhering to customs that cause hardship to people or lead to corruption or conflict is a reprehensible extremism and a form of excessive strictness (tanattu') forbidden by Islamic law.