Of course, but then you're not using horse archers but mounted infantry. And if you're using mounted infantry against a fort you are overspending, because they don't need the horses so they're just eating up your fodder for no purpose.
In other words, the horse archer isn't the be-all end-all of war, because in many cases they'll be better off not being a horse archer. Of course that doesn't mean the foot archer always beats the horse archer, either.
See, the thing about a mounted archer (or a mounted anything), is that an expensive (even that is debatable: most of the nomad population would own a horse and a bow just as part of their lifestyle) horseman who dismounts to fight on foot now and here, is much better than three cheaper infantrymen three hundred kilometres away. Tactically there are measures you can take against horse archers (same as against anything, really, nothing is the end-all of warfare), but strategically the horse archer is a fearsome and versatile tool and I'm surprised people keep repeating that it's easy to negate those advantages: it isn't.
Also, I've seen three separate claims in this thread that Batu suffered grevious casualties at Mohi and that the Hungarians were doing well. I know the English wiki says so, but what primary source is it relying on? I know that wiki policy is to rely on secondary sources for citations but I mean, I don't really care about those either. I need an actual period source with all these incredible details. I can't even find one. Seems like a bunch of just-so stories without citations with details that couldn't possibly be known, really. The first thing that Batu did after the aforementioned
grievous casualties was take and burn the Hungarian capital, and after that dispersed the army to raid the country, hardly the actions of someone in a precarious position.
Then he forced the Danube later that year. I mean, really.
Anyone have more knowledge of warfare during the crusades? The Crusader armies seem to have punched well above their weight against foes with a strong mounted archery tradition, supporting the argument that a skilled horse archer warrior class is not invincible.
Well, IMO you can summarise it thusly:
1. Quality of European armour and horses was a big shock to local horse archers, who weren't able to make a serious dent on the most capable parts of crusader armies in actual set battles.
2. The crusaders did well in sieges.
3. The crusaders ended up adopting horse archer auxiliaries anyway.
4. The crusaders were highly constrained in where it was safe for them to operate: they did poorly any reasonable distance away from the coast and being supplied by ships, partly because they had poor recon and poor logistics.