The Department of Mental Health shall investigate the case of each patient or trainee in a State mental health facility who is simply mentally or physically infirm or who is a harmless mental defective or harmless epileptic. When, in the opinion of the Department of Mental Health, the family, guardian, trustee, committee or other person legally responsible for the person is financially able to provide for his care, it shall, when in the opinion of the Department of Mental Health this is advisable, transfer the patient or trainee to the custody of that person. If all persons legally responsible for the patient or trainee are financially unable to provide for his care, the Department of Mental Health shall, when practicable, transfer the custody of the person to the county health authorities of the county of which the patient or trainee was a resident prior to admittance.