Is the prayer of a new congregant valid if he is led by someone who joined the prayer late, with a request to mention the books of the scholars that elaborated on this issue?

The prayer is invalid according to the Maliki school, the majority of the Hanbalis, and one opinion among the Shafi'is, for anyone who prays behind someone who is already praying behind an imam (a ma'mum) or behind someone who joined the prayer late (a masbuq). This is because a person praying behind an imam cannot be an imam himself, as his prayer is linked to that of his imam. This is the madhhab (school of thought) of Khalil in his مختصر (Mukhtasar - compendium), and it was explicitly stated by Ad-Dardir in his commentary.

The correct view according to the Shafi'is and some Hanbalis is the permissibility of praying behind a masbuq, because a masbuq is considered to be praying alone (like a munfarid) with regard to the part of the prayer he is making up. This is the view that is given fatwa (religious legal opinion) by some verifying scholars (muhaqqiqin), including Sheikh Ibn Baz, may Allah have mercy on him. They cite as evidence what is authentically reported from the Prophet, peace and blessings be upon him, that when he saw a man enter the mosque after the prayer had finished, he said: "Will not a man give charity to this one by praying with him?"