There were three mongolian armies. The one leaded by Batu, one that went through Silezia (Legnica), and the one (three units of Kadan) that invaded Transylwania. So Batu got enough men after Muhi whatever happened. They got enough infantry, too, mainly russians.

In 1242 Batu went to the Balkan from Hungary and stayed there until 1243. He didn't hurry back in the steppe.

Kills and casualties are not the same thing; Casualties are everyone that are not fit for duty after the action, including wounded, desserters/MIAs and so on. Actual dead at Kunersdorf where 6271 prussians and 7060 russian/austrians.
And while the casualty rate in this battle was quite extreme, that war generally saw very high casualty rates, frequently in the 20-30% range.
What made this war particularly bloody was probably a extreme level of discipline among the troops combined with the willingness of their generals to continue the engagements untill their armies where completely broken.

I do agree that 50% losses in a medevial battle is very high, and not very probable. However, the mongols where trying to take and defend a bridgehead with their mainly cavalry troops, and apparently this did not go very well for them untill their foes where flanked by another force that had crossed elsewhere.
As such, the probability that the force at the bridgehead, being unable to retreat in a viable manner, suffered heavy casualties is probable. But since the sources regarding the number of people involved in the first place are apparently hazy it is hard to say.

Suffice to say that in order to kill a very large portion of the enemy army, you need to have them exposed to efficient attack for a long time. Something that rarely happens unless the ground, general or doctrine forces people to stay in harms way.
Defending a bridgehead while avaiting relief is can be of these instances, though.

Mikael,

First off I am not saying there had to be 50% casualties. You are the one who keeps going back to that but I still think there is a strong amount of evidence for heavy Mongol casualties, maybe well up there 20 plus %. I will look into the Yuan Histories sometime to double check if I can find one in a language I can read but There are more than a few sources to indicate this happening and not just European literature. It seems to be a common theme for Mohi that the Mongols lost a large amount of men there.

It has happened in a number of medieval battles where they have huge casualties and though not the norm the things that make it happen tend to be the same. One being armies becoming trapped. I think most of the sources indicate Batu is trapped at the Sajo once he crosses, Bela finally gets the never to launch an attack, and as Elling said it is when the Mongol flanking force came the bad situation was reversed but the whole time before that Batu and his force were engaged in very hard fighting.

As you said the Mongols were already reaching their limits of resource.... so why think they could take over Europe. As well I think the Battle of Mohi and the stubborn defense of Hungary by her people was very much only exaggerating the problem in Europe for them. I do indeed think the fighting in Eastern Europe is responsible for turning the Mongol tide. It may not have been a string of victories but Fabian tactics often are far more effective in holding land. As long as the Mongols could not take the majority of fortresses and fortified towns they were on the losing sides of the war. They have some big actions there but most are raids. And I think the failure of the return of any to really push this area of Mongol conquest is also telling. The decade later would see Baghdad and the Sung dynasty fall so clearly the Mongols had some steam left but my guess is they were so divided even during Ogedai's reign but especially after and Europe clearly was not turning out to being a very easy conquest that they simply had no interest in a second round. So yes I still think what I thought before. Heavy casualties in Hungary, at Mohi and after contribute to them not wanting to return.

RPM

The Chinese used crossbows against nomad cavalry armies for over 1500 years. They considered it an effective counter.

So if you're interested in how crossbows worked against Xiongnu, Turks, Mongols and others like them check out Chinese military history.

This mid-thirteenth-century European source recommends bows and crossbows for fighting against the Mongols. In the fifteenth century, Bertrandon de la Broquière gave rather similar advice for combating the Ottoman army.

In the well known memoirs of Giovanni Di Plano Carpini, who traveled under Papal directions all way back to Great Khan into Mongolia in 1252, Historia Mongalorum Quo S Nos Tartaros Appellamus, The Story of the Mongols Whom We Call the Tartars, while giving some tactical information against Mongols, he mentions crossbows;

Whoever wishes to fight the Tartars should have these weapons: a good bow or strong crossbow (which they fear), and enough arrows and a good axe of good iron or a hatchet with a long handle. The points of the arrows for the bow or crossbow should be tempered when they are hot in water mixed with salt, as the Tartars do, so that they should be very strong for penetrating their armor; swords and lances with a hook, which are good for pulling them from the saddle because they fall easily from it, daggers, thick cuirasses because arrows do not easily penetrate these, and a helmet and armor and other things to protect the body and the horse from their weapons and arrows. And if some are not as well armed as we have described, they should stay behind the others, as the Tartars do, and shoot against them with bows and crossbows.

He mentioned about crossbows and added that "which they fear" but there is no explanation why they fear but in this case in could be speculated that Mongols experienced the deadly hail of bolts so they are more cautious about his weapon.

it also mentions them falling easily from there saddles which is simply insane considering their skill as horsemen. not saying that they didn't respect the effectiveness of crossbows but that we should take his view with a gain of salt and that he may have had a bias also the fact that the Chinese used crossbows against horsemen might i remind everyone that the yuan dynasty was Mongol

R.,

In the context I think it makes sense. Good at riding horses they may be but if I pull on you with a iron hook you are likely coming down.