Is it permissible for the muezzin to make the call to prayer and then leave to pray in another nearby mosque, while the imam remains in the mosque to lead the prayer, and are his previous salaries considered lawful?

The employee must fulfill his job duties. Whoever falls short in this regard is considered negligent and is deprived of his salary to the extent of his negligence, for Allah the Almighty says: "O you who have believed, fulfill [all] contracts," and "Indeed, Allah commands you to render trusts to whom they are due," and "Do not betray Allah and the Messenger or betray your trusts," and "Do not consume one another's wealth unjustly." If the adhan (call to prayer) and iqamah (call to commence prayer) are among the duties of the muezzin, then it is not permissible for him to leave after the adhan. The general rule is that a worshipper should not leave the mosque after the adhan for a prayer without an excuse, due to the saying of Abu Hurayrah (may Allah be pleased with him): "As for this one, he has disobeyed Abu al-Qasim (the Prophet, peace be upon him)" regarding one who leaves the mosque after the adhan. And the saying of Al-Tirmidhi: "No one should leave the mosque after the adhan except for an excuse." Therefore, if the muezzin's custom is to leave the mosque after the adhan to perform the prayer in another mosque, and his duties include the adhan and iqamah, then this is not permissible. He should let someone else make the adhan if staying becomes difficult for him. The amount he received corresponding to his negligence should be estimated and given to the poor and needy.