What are the jurisprudential rulings that permit a father to pay Zakat al-Fitr on behalf of his adult children without their permission, and is it permissible for the one paying Zakat to not pay Zakat al-Fitr on behalf of his father, to whom he provides financial assistance?

Zakat is an act of worship that requires intention from its giver. Therefore, giving Zakat on behalf of another person is not valid unless it is done with their intention, except in the case where a man gives Zakat al-Fitr on behalf of his adult, poor children whose maintenance is obligatory upon him. In this case, it is permissible for the man to give Zakat on their behalf without their permission because their maintenance is incumbent upon him. However, if the father is self-sufficient, then the Fitr Zakat for himself is obligatory, and the son is not obligated to pay his father's debt.