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.
There is difference between a standard horse archer (just out of my memories, vastly used by hungarian, austrian, bohemian and polish armies until about the end of the 16.century - at least in the case of Hungary), the mongolian horse archer (whom i dont like to call horse archers, because this leads to misconceptions, they were more like "like knights or samurai trained in their whole life for combat with a little bit less armor and excellent mounted bow training professional warriors") and the "internet horse archer", the mythical creature and never existed.1
Neither was really fearsome - that was the heavy shock cavalry!
And they were not really versatile either, since usually they lacked the armor, tended to be outranged by leg bowmen (crossbowmen in the siege of Esztegrom outranged the mongolian dismounted archers - okay, it was a siege, hilltop, walls, whatever, but still, but again, at the Battle of Muhi crossbowmen in open field outranged them so...) and against well armored (or shielded) foe they simply lacked the punch (everyone lacked the punch except crossbowmen and firearms.)
I would join the "not the weapons, but the men" flock in case of the mongolians: tactical, operational, command and control and discipline superiority over the western armies. Once they started to lose these, they got degraded to the "raider" category.
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.
There are very few primary sources about the battle and the immediate aftermath (Rogerius, an another dude whom name i forgot and some of the letters of Béla) and they are full of shit.
If you are cross-examine them with secondary sources and the events happened around the time, things dont check out.
For the girevous casualities: well, of course we dont have the number and pretty much no one (important) survived the action from the hungarian part, but the battle raged for about a night (and in the next morning) at the bridge and the detached forces returned to the camp victorious (and started to party a little). This, some non-western sources mentioning high casualities and the fact, that Subotai changed plans for the battle overnight indicating heavy casualities, most likely among the press-ganged auxiliary forces but at least one high-ranked mongolian lieutant (and his unit) mentioned by name as fallen, so possibly among the core mongolian too.
Next morning fightings were also heavy and after the flank attack, the endgame at the camp lasted till nightfall, but that was mostly mop up.
So, again, grievous? We dont know. But heavy for sure.
But it did not matter at the end, because after two failed tricks, Subotai could had make a third and a fourth - both succesful.
(Yes, im imoressed with him.)
And Batu did not took the haungarian capital - at that time there were two: Székesfehérvár (failed siege) and Esztergom (city taken and plundered, castle held). They did not forced the Danube, the river frozen solid and they crossed it unopposed (and after it, they captured the city of Esztergom, plunder etc.)
Oh, and while the hungarian army pretty much destroyed (despite the really grievous casualities, not really), the mongolian army did not. They raided their path - and they excelled at it -, but the armies never dispersed. They were not fools.
