N.Y. Comp. Codes R. & Regs. tit. 18, § 311.4
(a) In accordance with paragraph (d) of subdivision 5 of section 62 of the Social Services Law, if a town, city or county provides care for a person in a family home, boarding home, intermediate care facility, nursing home, hospital or institution outside of its territory and pays the vendor or supplier for such care, or sends a grant to the recipient, the agency making such provision shall continue to be responsible for payment for such care as long as the recipient of assistance or care is in need thereof. In the event that 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 agency which would be responsible for such required assistance and care if such person had remained in the territory of the town, city or county making such provision.
(10) Under the provisions of an amendment to paragraph (c) of subdivision 5 of section 62 of the Social Services Law, a new paragraph was added which relates to the termination of continuing liability as to children. This revision is incorporated in chapter 41 of the Laws of 1958. Thereunder, when a child who has been cared for away from his home by or on behalf of a public welfare district, pursuant to title 2 of article 6 or other provisions of the Social Services Law, the Children's Court Act of the State of New York, the Domestic Relations Court Act of the City 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, this formal discharge shall terminate the responsibility of such public welfare district to furnish assistance and care for such child; following such a 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 such assistance and care for him as for any other child residing in the district.