Is the man responsible for his children if he divorces his wife and they live with her?
Yes he is responsible for them even if they don’t live with him due to the fact they are his children. He should financially support them, because the Prophet, alayhi salaatu was salaam, said in a hadeeth reported by al-Imam Nisaai and others:
كفى بالمرء إثماً أن يضيع من يقوت او يعول
‘It is sufficient as a sin upon a man that he does not take responsibility over those he is responsible for.”
A Side Note: Although the origin of this hadeeth is in muslim with the number 996.
However it is only obligatory upon him according to his capability, as Allah mentioned in the chapter talaaq (divorce) which the meaning is:
“Let the rich man spend according to his means, and the man whose resources are restricted, let him spend according to what Allah has given him. Allah puts no burden on any person beyond what He has given him…”
~ Abu Fajr AbdulFattaah bin Uthman