What is the ruling on giving out Zakat al-Fitr on behalf of a father performing 'Umrah in Saudi Arabia at its appointed time, if the giving out is done by the children in Egypt one day after the 'Eid prayer?

The basic principle is that Zakat al-Fitr should be given in the country where it became due. However, it is permissible to appoint someone to distribute it in another country if there is a compelling interest. An expatriate should give Zakat al-Fitr for himself in the place of his expatriation. If he cannot find deserving recipients there, he may authorize his family to distribute it on his behalf in his home country. It is not permissible to delay Zakat al-Fitr beyond the Eid prayer, except for a valid excuse. The time of obligation is what determines when it should be given, and forgetfulness is an excuse that absolves one of responsibility. The delay of Zakat al-Fitr by your father beyond its specified time is not permissible in Islamic law, and repentance for this negligence is obligatory, unless there was an excuse. If it has been paid to the poor, then the obligation to pay it has been fulfilled.