Is it permissible for a father to pay Zakat al-Fitr on behalf of his financially independent son?

A capable son pays his own Zakat. If his father also pays it on his behalf, there is no harm in that, especially if the father is accustomed to it, as he might be pleased to continue this custom. The son can let his father pay it on his behalf and also pay his own Zakat. Some consider the father's continued payment of Zakat al-Fitr for his children as a symbol of their connection to him and their remaining under his obedience and care.