For a lot of specimens exist, from both era I wrote about in „Building up a character”, I will concentrate on usual items. I made two lists of few items, one for a 13th cent. and one for a 16th cent. Mamluk cavalryman. You can find many pictures of period equipment searching the web but try to find such photos from museums, on which you can see the dating and origin.
Though I am sure real warriors sometimes used not typical or very old equipment (like heritage) I recommend to use only typical items which perfectly fit your chosen era. There are too few serious reenactors of the Middle-east to portray rare exceptions instead of „regular” fighters.
Even museums can make errors, but thoose are hard to notice. For example a sabre-blade was forged in 1500, it is written on the blade, but its hilt was remade in 1700 with a totally different form. Then in the museum it is displayed as a saber from 1500, because it is „evident”, it is written on it. A reenactor orders a replica of it and shows the museum picture that it is correct for 1500, it is just a „rare” saber, or the „first appereance” of that type. (and he may be right:))
There are much more obvious errors. I don’t know why but museums like to put the greaves instead of the bazubands on the lower arms of display figures. Well, its broadening part which protects the ankle seems to be good for the hand, but if it is put to action it would make the wrist unmoveable.
Therefore the safest way is to portray usual equipment which is 100% fit for the era and its use is evident.