My speciality is the Crusades, so my recommendations all approach the subject from that area. There's a lot to be learned about medieval Islamic culture that isn't filtered through this lens, but unfortunately I'm not qualified to recommend any books in that area.

If you're looking for a deep dive, Carole Hillenbrand's massive The Crusades: Islamic Perspectives is the single-volume history of the subject. It covers a lot more than what you're asking, since it spans the full Crusading period, but it definitely provides the kind of deep analysis you're looking for and it's references are full of more detailed books and articles should you wish to dig even deeper.

Francesco Gabrieli's Arab Historians of the Crusades also deserves a mention. This is a collection of primary source documents (in translation) written by primarily Muslim authors and providing their perspective on the events of the time. Like Hillenbrand's book, this covers the full Crusading period, so it's not as focused as you might like, but it's a great entry point if you're looking for the sources themselves.

Lastly, Anne Marie Eddé's amazing biography Saladin is a great place to get exactly the kind of details you're looking for, albeit about the founder of the Ayyubid dynasty rather than his descendants. That said, Eddé does include information on where the dynasty went after Saladin, and his family and children factor into the story, so it's not without information on the later dynasty.

Unfortunately, I don't really know enough about Mamluk historiography to recommend anything specific on them.