(b) If a public welfare district, town or city provides care for a person in a family home, boarding home, nursing home, convalescent home, hospital or institution outside of its territory and pays for such care directly or through a grant made to the recipient, the public welfare district, town or city making such provision shall continue to be responsible for payment for such care as long as the recipient is in need thereof. In the event any other type of public assistance and care is needed by a person receiving such care, it shall be furnished and paid for by the public welfare district, town or city which would be responsible for such required assistance and care if such person had remained in the territory of the district, town or city making such provision; the public welfare district, town or city making such provision shall likewise be responsible for the care, removal and burial of the body of any such person who shall die, and the expense thereof.
The public welfare district, town or city which, pursuant to the provisions of this paragraph, is providing public assistance and care for the mother of an infant, on the date of birth of such infant, shall be responsible for providing public assistance or care required by such infant on and after the date of his birth; and such public welfare district, town or city shall be deemed to have made provision for the care of such infant outside of its territory.
When a child who has been cared for away from his own home by or on behalf of a public welfare district, pursuant to title two of article six or other provisions of this chapter, the family court act of the state of New York, or other provisions of law, is discharged, pursuant to law, to his parents or parent, brother, sister, uncle, aunt or legal guardian, who are or is then residing outside the territory of such public welfare district, such discharge shall terminate the responsibility of such public welfare district to furnish public assistance and care for such child pursuant to this paragraph; and upon such discharge it shall become the responsibility of the public welfare district wherein such child is thereafter to reside with his relative or legal guardian to provide necessary public assistance and care for him as in the case of any other child residing therein.
The provisions of this paragraph shall not be deemed to authorize or empower towns or cities of a county public welfare district to exercise responsibilities with relation to public assistance and care inconsistent with the responsibilities imposed or conferred on them by other provisions of this chapter.