Does the father have to pay zakat al-fitr on behalf of his children if they live with their mother?

Firstly:

Zakat al-Fitr is an act of worship that the Muslim is obliged to perform by himself or through his deputy or proxy. This has been discussed in the answer to question no. 99353

Based on that, the husband is not obliged to pay zakat al-fitr on behalf of his wife or his parents. But in the case of his children, if they are adults of sound mind, he does not have to pay zakat al-fitr on their behalf either, whether they are rich or poor. But if they are still minors, if they have their own wealth then they should pay zakat al-fitr from their wealth; if they do not have any wealth then zakat al-fitr should be paid on their behalf by their father, even if they are in the mother’s custody.

Al-Nawawi (may Allah have mercy on him) said:

If a child does not have any wealth, his zakat al-fitr must be paid by his father, according to scholarly consensus, as was reported by Ibn al-Mundhir and others. If the child has wealth, then he must pay his own zakat al-fitr. This was the view of Abu Haneefah, Ahmad, Ishaaq and Abu Thawr.

End quote from al-Majmoo‘, 6/108

He also said (6/77):

If the child is well off, then his maintenance and his zakat al-fitr should come from his wealth and not from his father or grandfather. This was the view of Abu Haneefah, Muhammad, Ahmad, and Ishaaq. Ibn al-Mundhir narrated from some of the scholars that it should be paid by the father and if he pays it from the child’s wealth, then he has sinned and is liable (i.e., he has to pay it back). End quote.

Once this is established, with regard to the ruling that zakat al-fitr must be paid by the father for small children, it makes no difference whether they are in the custody of their mother or under the care of someone else, because he is obliged to spend on maintenance, if they have no wealth, according to scholarly consensus, and the same applies to zakat al-fitr.

Ibn al-Mundhir (may Allah have mercy on him) said: All the scholars from whom we learned knowledge are agreed that a man is obliged to spend on his children’s maintenance if the children have no wealth.

End quote from al-Mughni, 8/169.

For more information please see the answer the question no. 111811.

And Allah knows best.