The association between Daoist/Buddhist monks and various Chinese martial arts is pretty well known. It would seem strange for them to be unique in this connection between mysticism and martial arts.
Of course there's the famous Assassins, but what were their martial arts/fighting styles like? Were there other militant groups like them, perhaps associated with Sufism?
And even earlier--what about the Zoroastrians and zurkhaneh?
The association between Daoist/Buddhist monks and various Chinese martial arts is pretty well known.
It's well-known, but largely derived from fiction. In particular, Daoist monks have little, if any, connection with martial arts. The histories crediting Daoist monks with the creation of various internal martial arts (e.g., Taiji) are fictional. The "Daoist" origin stories were written at a time when Shaolin was famous as a source of martial arts (more on this below) and those seeking to publicise their own schools/styles would claim their martial arts descended from Shaolin (even if entirely unconnected). The versions where legendary Daoist monks from centuries earlier are the "inventors" of the arts come from the Qing Dynasty, and use "Daoist = real Chinese", "Buddhist = foreign = Manchu" for their advertising.
Shaolin Temple became known for martial arts in the late Ming. Before this, some other temples were occasionally mentioned in connection with martial arts, but there was no major connection with martial arts. The rise of Shaolin in the martial arts imagination was driven by two things: firstly, successful participation in anti-pirate battles by Shaolin monks (and monks from other temples, too, but they didn't get the same popular credit), and secondly, Ming fiction linking martial arts and monks. The modern connection between monks and martial arts is strongly driven by movies; the Ming connection was strongly driven by the Ming version of the martial arts movie: the martial arts novel.
As noted above, the Shaolin-martial arts link does have some basis in fact. I wouldn't call it a link between mysticism or religion and martial arts, but a link between monasteries and martial arts. Why are monasteries linked to martial arts?
Monasteries are often major land-owners. As such, they often raised their own armed forces. For example, the first reliable historical of Shaolin monastery connected with violence is Shaolin raising a private army to recover some land taken from them by a local warlord in the 7th century. Monastic armies also appear in Japan and and Western Europe, used for similar purposes: to protect monastic lands. Where private armies are maintained, private armies are often trained: thus, martial arts.
Monasteries often have many young men, and older men, with an interest in fighting and/or experience in fighting. For example, warriors might retire to monasteries (a common element of martial arts novels/movies, but it also happened in real life). Even if such men are not part of an official monastic army, but are nominally monks (even if they don't live under monastic discipline, or follow monastic rules), there can act as a source of armed force.
When monasteries play the game of civil war politics, assisting one side in the hopes of post-victory reward, they can become well-known (at least for a while), and obtain legal and financial benefits. If their chosen side loses, they can be destroyed. Win or lose, this type of activity does make monasteries more visible in a military context. Helping the established state against foreign powers can also work, at lower risk. Shaolin played this game, with some success. Japanese monasteries played a similar game, leading to Oda Nobunaga's destruction of the monasteries and their monastic armies on Mount Hiei in 1571. Korean monks obtained concessions from the Korean government for their fighting in the Imjin War. In 8th century Ireland, monastic armies fought each other to further their ambitions, as well as supporting secular powers in their wars.
There is a broader connection between mysticism and religion and fighting.
Those who fight often use mysticism and religion to support their fighting. Fighting in earnest is very stressful, and methods to reduce stress in combat can have real practical benefit. Thus, fighters will use protective tattoos, protective amulets, protective incantations (sometimes all at once!), etc. during fights, and meditative practices as part of their training.
Some will fight as part of their perceived religious duty. This can be promoted by the state in order to recruit and motivate soldiers, and/or can be bottom-up, with sincere individuals becoming crusaders, jihadists, juramentados, etc. This can also be combined with monastic armies in the service of supporting monastic power, with consequences such as the conquest and rule of Prussia and Livonia by monastic military orders.
But as for the question in the OP, Islamic monastic martial arts, I have very little to say. As for Assassin martial arts: nothing special, with their regular military forces fighting with the usual weapons of the area in the same way as everybody else, and assassins depending on their willingness to die rather than any great martial arts skill (stabbing their target repeatedly until the victim's bodyguards kill them is an unsophisticated technique depending on surprise). But the same was true for most monastic armed forces: the same weapons as usually used, used in the same way as usual.