Hello,

While there is nothing similar to tournaments in the Islamic world, in Central Asia, Iran and in India a similar concept did indeed exist. This was the Royal Hunt. Here men, recruited by the nobles, the nobles themselves and the Khan and Khagans used to practice their horseback archery while hunting various animals or even, in the case of India, hunting on Elephant backs with gunpowder weapons.

The best resource to look at is The Royal Hunt in Eurasian Society by Thomas T. Allsen which talks about the usage of royal hunts both as political and militaristic means of projecting power and The Tarikh-i-Rashidi by Mirza Muhammad Haydar Dughlat translated by either N. Elias and D. Ross or Thackston where Royal Hunts are depicted in an actual historical setting.

The Muslim world equivalent to a tournament would have been a "furusiyah" tournament, from the Arabic "faras" (meaning "horse", possibly derived from the sound of a horse's hooves smashing the ground): compare "faris", meaning a mounted warrior.

There is an extensive body of medieval furisiyah literature. The most well developed sources appear to date from the Mamluk sultanate, which controlled Egypt and parts of the Levant for centuries: however, see Shibab al-Sarraf's "Mamluk Furusiyah literature and its antecedents", which argues that Mamluk sources borrow extensively from earlier Ayyubid, Abbasid, and even Umayyad traditions.

The word more generally refers to the body of knowledge a mounted warrior would have been expected to demonstrate, such as how to care for a horse, charging with a lance, archery on horseback and on foot, the use of the sword and mace, military strategy, and of course polo (no, really). It can also refer to the military ethos in general, comprising "bravery, manliness, gallantry, and generosity" (compare the European sense of "chivalry"), and came to be used interchangeably with "futuwah", a more general sense of code of honor for warriors (from "fata" meaning "young man").

The science of "furusiyah" appears to have reached its apex during the Mamluk sultanate that controlled Egypt and parts of the Levant, and grew naturally out of the skills mounted warriors would have been expected to exhibit. Unfortunately I am having difficulty finding information on what exactly a furusiyah tournament, if it happened, would have entailed: this is compounded somewhat by the fact that "exercise" and "game" are also used to refer to Mamluk equestrian training in general, not necessarily the sense of an organized competition. Rabie's "The Training of the Mamluk Faris" (hosted here, see here for a version with sources) comments on some of the specific training exercises involved.

From what I can tell, Ayalon's "Notes on the Furusiyya Exercises and Games in the Mamluk Sultanate" (collected in Ayalon's "The Mamluk Military Society") would be a good source, but unfortunately I can't seem to track down a copy (it's also from the '60s, reprinted in '79, and may be somewhat outdated).