By Attila the Huns had largely settled down and fought as infantry. Besides, the Big A didn't have to work his way through endless webs of fortifications specifically designed to grind down the momentum of invaders.
Yes I am well aware that the Huns adopted infantry in their armies, it was the point I was making. Something similar happened in China, where the Mongol armies conducted seige after seige not feigned retreat with horse archers.
As for pasture, the Mamluks made a point of burning the grasslands of Syria and destroying or appropriating the local granaries which duly caused the Mongols fairly severe logistical issues. Go fig.
The Mongols succeeded in taking Syria. Logistics was a problem more so because of the constant Mongol threat posed by the Golden Horde and Qaidu. They failed to keep Syria because they were too strategically stretched.

I strongly suspect the Mongols abandoned their Hungarian aquisitions and retreated back to the steppes partly because they had amassed enough intel on Europe to decide the poor, backwards sub-continent chock full of forts and highly territorial, xenophobic bastards just plain wouldn't be worth the trouble to try to take over. Most likely they also noticed they had run out of steppe, and if they were going to keep going and hold their new aquisitions they'd be forced to abandon the nomadic life - the same, after all, had happened to the Hungarians only a few hundred years earlier, and I'd be very surprised if the Mongols didn't pick that detail up at some point from their new subjects.
With one of the finest intelligence systems I doubt they suddenly discovered any of these things. We all know that Ogodei died and we also know that the majority of Mongol contingents also returned for the Quriltai. Nomadic life was not maintained in China either.

Then there's also the little fact they'd suffered comparatively high losses in that famous river battle against the Hungarians and Templars when trying to force a bridge crossing in the face of astonishingly small number of knights (I've read the night-time attempts were repulsed almost entirely by just the bodyguards of the Hungarian King and the Templar Grand Master - that more troops could not be thrown into the fray, and that the crossing attempt was noticed purely by luck, incidentally tells something of the degree of professionalism and discipline involved...). If they were at all informed of the geography ahead it is perfectly conceivable they weren't one bit happy about the prospect of having to fight over several similar chokepoints for the fairly meager gains Europe promised, nevermind now the projected logistical problems.
Sure, they did suffer heavy losses at Sajo, not so much while re-taking the bridge but rather, after they had succeeded and when they were now outnumbered and hemmed in with the river at their backs. The second crossing being found purely by luck is somewhat fanciful to say the least, made even more so when we consider it was they and not the Hungarians who chose the battlefield. What is more, I hardly see Subedei relying on chance and we have an account of the day at a celebratory banquet, where Batu was reminded by Subedei that he should have delayed his first assault knowing that timing was crucial. Hardly the words of a general who had just been blessed by good fortune

......Orda