Skirmish tactics were typically relegated to Turkish or other steppe-nomad mercenaries those days anyway. They had the practical know-how for it, as well as the requisite herds of ponies for remounting after all the running around tired the beasts. Mind you, steppe skirmishers were always only too happy to don a shirt of mail or a lamellar corselet if they could get them too, and helmets were ubiquitous.
Having something more than a shirt and your rather frail human skin between your innards and the enemy's assorted pointy things sort of appeals to most sensible people. And the nomads were by and large very sensible and pragmatic folks.
The horse-archers of more sedentary nations - Arabs, Byzantines, the Mamluks of Egypt - normally fought in close order and didn't skirmish; they didn't have the remount pool to replace the tired horses the harassing tactics invariably produced, and thus relied on density and weight of fire and cohesive squadrons instead.
Anyway, Middle Eastern shock cavalry was usually sort of "dual purpose" - many doubled as mounted archers - and favoured hit-and-run tactics of controlled charges and withdrawals repeated until the enemy broke. This had the plus side of preserving great degree of tactical maneuverability. Conversely their European colleagues were pure shock specialists trained to (hopefully) smash the enemy formation with a single massed charge with couched lances (a spear-wielding technique, incidentally, which the Easterners also knew and used; the Arabs called it "Syrian attack"). They were in a sense giant battlefield projectiles, and their emphasis on linear attack tactics left them somewhat unwieldy. This duly led to the peculiar battlefield dynamic where the Muslims tried to goad the Franks (as the Europeans were generally called) into committing the charge inefficiently, and the Franks strove to hold it until a suitable tactical opportunity presented itself. A natural side effect was that the armies of the Crusader Kingdoms were by far more disciplined and controllable than their peers back in Europe, and had rather better honed practices of combined infantry and cavalry operations (since the crossbowmen and armoured spearmen were vital in keeping the Muslim horse-archers from slaughtering the knights' horses and generally holding them at an arm's reach).
The warriors of Outremer also tended to be ever frustrated by Crusading newcomers fresh from Europe who had no idea of how war was waged there. I've read one reason for example Richard Lionheart employed local troops as the outer guard of march columns was that they could be counted on to not do something really stupid like going off to chase after skirmishers...
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