Saladin's horse archers is more likely than not lightly armoured, but they are also coming from horse-owning part of their society, and quite capable to fight on foot as light infantry if required. While I must say I didn't know if they carry two bows each (one is mounted bows and one larger infantry bows), I am quite certain that they all carry swords and slung a shield on their backs, if the terrain did not favor mobile horse archery tactics, they will dismount and fire on foot.
And yes, quite unlike the European horse archers who fight on foot, Middle-eastern horse archers are trained to fire their bows on horseback. If the terrain is open and conductive to cavalry warfare, I must say they will stay mounted and outflank their longbow armed enemies. You may say that outranging the horse archers will defeat them, but approaching them from multiple directions is still a better tactics, yes, the horse archers will sustain more initial casualities, but once the longbowmen are encircled, they're gonna end up dead anyway. And we must know that their ammos aren't unlimited, longbowmen can only carry 24 - 32 bodkin arrows in their quiver, and uh, they aren't gonna carrying more than two quivers, while middle eastern arrows is far lighter and they can carry 30 - 40 arrows in their quiver, and since they're mounted, they can carry at least three quivers. While the bodkin arrows has more penetrating power and accuracy, the sheer volume of the lighter arrows will saturate the poor, unarmoured longbowmen.
And let's remember this, the yeomen archers in the English society aren't as well off as the Horse archers, they are quite likely only has helmets, and mauls. While this can be deadly against heavily armoured French Knights, who're encumbered by armour, this time they aren't fighting heavy infantry, but light infantry armed with swords (the Saladin's dismounted horse archers). Heavy weapons are good vs slow opponents, but against fast, agile opponents, they're pretty much done for.
But yeah, if the English heavy infantry are counted, compared to Saladin's levied Arab spearmen, they will be the most descisive part of the battle that will reverse the weakness of the longbowmen. Saladin's levies are little more than arrow fodders, and can't do much in combat, while the English heavy infantry are heavily armoured, and is more likely than not capable to soak up the horse archers' arrows without sustaining much casualities. The only way for the horse archers to inflict damages against those heavy infantry is by using close range bow tactics to provide better penetration, but the english longbowmen will surely neutralize them well.