J - PSD is correct. The military sponsor must request that the family members be allowed to stay once the member is PCS'd. The most common reason is in order to complete the school year. This is a formal request sent from the sponsor's command to the Naples Base CO. If granted, the family would be allowed to stay in current quarters (military/offbase). Keep in mind, this is not automatic. The Base CO is under no obligation to approve it. Those that are approved generally are within 90 days of the end of the school year/end of semester.
A command will never sponsor a family to remain in area and collect TLA/Per Diem...not sure where you may have heard that, but it is not at all true and would be in violation of federal regulations.