How do the latecomers who joined the prayer, pray when their Imam leaves and they are put forward to complete the prayer, while they were originally latecomers?

If the Imam is prevented from completing his prayer, he is permitted to appoint a successor, and this is the correct view among the sayings of the scholars.

If a latecomer (masbuq) is appointed as successor, this latecomer builds upon his own prayer, not upon what the first Imam prayed. As for the congregants, they have three options: 1. The latecomer appoints someone to lead them in salutation (tasleem), then the latecomer stands to complete the rest of his prayer. 2. They make the salutation individually if their prayer is complete. 3. They wait for the latecomer to complete his prayer, then he leads them in salutation. This last option is the most preferable.

The jurisprudential schools have differed on the details of a latecomer's succession and how he and the congregants should complete the prayer: Hanafis: It is preferable not to appoint a latecomer as successor. If he is appointed, he starts from where the Imam finished and leads those who were present from the beginning (mudrik) in salutation. Malikis: It is stipulated for the successor that he must have caught a significant part of the rak'ah. If a latecomer is appointed, he prays with them according to the system of the first Imam's prayer, then he signals to them to sit while he completes his prayer, then he makes the salutation with them. Shafi'is: The succession of a latecomer or anyone else is valid. If the latecomer completes the Imam's prayer with the people, he then stands to make up what he missed. The congregants have the choice between leaving him and making the salutation, or waiting for him while sitting to make the salutation with him. Hanbalis: The succession of a latecomer is permissible. He builds upon what has passed of the Imam's prayer. If the congregants finish their prayer, they wait for him until he completes it and makes the salutation with them. If they make the salutation without waiting for him, it is permissible.