Earlier quoted context omitted.
It's hard to compare Mongols to Europeans. The composite bows rely on animal-based adhesives, which are great in cold and dry climates, but fall apart in humid and warmer climates. Also, light cavalry armies are useless against castles and fortified cities as the ones found in most of Europe, specially when you don't have huge, flat battlefields that allow light cavalry to circle any target. In other words, Mongol wa…
Re: "light cavalry armies are useless against castles and fortified cities as the ones found in most of Europe" Not true, as the Chinese cities at the time were much more fortified than Europe. The Mongols were able to plunder the surrounding lands. Also, despite Europe's fortifications, a scouting party led by Subutai of 20,000 horse archers, widely wiped out a much more significant force of Central Europe soldiers,…
European heavy cavalry was not useless against mongols. It is just that they were seldom effective when mongols could just run away from the battle field. Easy to do when they were much lighter armoured. But if Mongols maintained a siege they had to stay in position which would have made them vulnerable to European nights which were probably superior to mongol soldiers if they could not perform hit and run tactics.
Apparently the mongols beat the Hungarians initially but when they came back and the Hungarians had built lots of castles, they were soundly beaten. The sieges dragged on and they would get harassed by Hungarian forces which could always retreat back to castles while mongols starved.
My understanding is that the Mongols managed to conquer the more hilly southern China by using conscripted Chinese soldiers. So they were not really using standard Mongolian tactics there. Duplicating this in Europe would have been difficult because unlike China Europe did not have the same kind of central government which would have allow the conquest of a few cities to gain huge tracts of land and extra manpower. They would have had to conquer huge numbers of castles which would have dragged out in time.
In fact Machiavelli talks about this in the Prince. The Ottomans, Roman Empire, China etc were far more centrally controlled which made them one strong unit but once you conquered the central city of government you would control the whole country. The Feudalistic European countries however might not as easily marshal huge combined forces but they were extremely difficult to conquer because you had to conquer every little vassal state one by one. Considering that every one had a castle this was no easy task.
One should not completely discount the effectiveness of European military. During the crusades, the muslim forces would usually use Mongol like tactics to wear them out. But whenever the crusaders managed to corner a muslim army they crushed it with their heavily armoured knights.