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

Firstly: 

Zakaat 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 zakaat
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 zakaat
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 zakaat
al-fitr from their wealth; if they do not have any wealth then zakaat
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 zakaat 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
zakaat 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 zakaat
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
zakaat 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 zakaat
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.