Seems I was mistaken. Still, the fact remains that you can probably have 2-3 times as many foot archers in the same space as 1 horse archer,
so even if there's an equivalency of power for 1 bow it's much easier to fire the longbow en masse rather than picking out individual targets.
Most of the mass shooting both in the case of the English archers and in the case of the Mongols was a barrage fire at a high angle and not a precision fire (for which Genghis described special tactical form, "a chisel", used by a single rank of the skilled archers). Density considerations are not very important in both cases because a lesser density on nomadic side is being compensated by a speed of movement. Anyway, the whole comparison is pointless: the foot archers could not be used without a proper backup by a "heavy" infantry and in a battle they are pretty much static and, as a result vulnerable. Look what happened to the Janissary (very good archers with the powerful bows) at Ankara: their formation was surrounded and they were massacred.
While it seems the max range for some mongol bows was ~500m compared to ~400m for an English bow, the mongols' figures were highly extraordinary. I couldn't find the average mongol horseman's max range, but ~250m seems to be a consensus for English longbows. Thus, assuming that the Mongols would have to close to within ~250m themselves to provide anything more than harassing fire, the logic of spacing would give the advantage once again to the English.
All these, quite popular, numbers are a result of a profound confusion between a notion of a maximum range of a weapon (usually in a target practice), an effective range of the same weapon when used by the big masses in the battle conditions and the killing range at which the target (especially armored) could be seriously hurt. In other words, pretty much irrelevant.
If they could fire effectively at longer ranges than that, I'd imagine that field fortifications would still have some effect if they could be forced to come to such range as to give plunging fire.
The whole purpose of the fortifications was to provide cover for their defenders and, except for some Chinese "historic" movies, the fortifications had been rarely taken just by shooting the arrows: the attackers had to dismount and proceed with the boring tasks of climbing the ladders, breaking the gates, etc. In other words, get close enough to the walls and, quite often to put themselves in a position where they simply can't use their bows. Without any doubts the Mongolian bows had a higher range than the Russian circa XIII and yet, the cities with rather primitive fortifications had been taken by storm and not by an archery barrage which means that even a stockade can provide a considerable protection against the archery.
Furthermore, it seems to me that the composite bows in use were much more fragile than the English longbow and any moisture could seriously damage the glue holding them together, compared to the English whose only concern was a wet bowstring. All this aside, given the standard skirmishing tactics of horse archers, I think that longbows especially could well be used to devastating effect, especially with broadheads.
Finally, here we go with the "moisture" mantra (it did not take as long as I hoped)!
To start with, "classic" Mongolian bows had been using the fish glue with stands up to the moisture quite well. Second, an idea that the nomads (in general) would become weaponless after a single rain is something worthy of Lord of the Rings: just wait until it starts raining and the weaponless enemy is running away. Third, if you started with the Mongolian bows, then it makes sense to talk about the Mongolian tactics and the Mongols had a powerful armored cavalry, not just the light mounted archers. Fourth, it does not really matter to you if you are being killed just because your bowstring is got wet and you are left with just a stick in your hand while being run down by a cavalry armed with the spears and swords (and holding their bows safe in the leather bags).
English archery experience was against a substantially different opponent: much slower, incapable of maneuvering, and quite often fighting on foot (the French started dismounting their knights relatively early in the 100YW). Even then, success required a reliable defensive position from which they could do shooting and the enemy slowly coming at the front. Now, what would they do with the enemy who is bypassing them on one or both flanks, which was one of the standard Mongolian and Tatars tactics? What would they do to the enemy who, following Pratchett's advice (

), just allows them to
stay in a good position (you can check what happened to the Kievan troops at Kalka)? The foot archers can't maneuver on a battlefield, they can't pursue a mounted opponent and they can't ran away from him.
The broadheads are fine but you have to hit someone with them which is not simple is you are shooting at the fast riding cavalry. And with the standard English formations and barrage fire the effect would be rather unimpressive. Actually, it was not very impressive even in OTL unless the shooting was done at point blank range as at Crecy or French cavalry detachment at Poitiers.
Oh, and one thing I'd add to the list of anti-horse-archer measures would be light cavalry mounted on camels instead of horses, where practicable, given that camels have been known to spook horses.
The best scenario would be the English archers riding the camels and elephants: all the local horses would be spooked at no time.
Can you please explain why the horses brought up in an environment where the camels had been a standard part of a livestock would be spooked of them?