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.

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.

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.

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.