The Mongols never made the gunpowder leap, but what if they had? Muskets were probably not suitable for their horse archery way of war, but what about horse artillery?
Would being able to rapidly maneuver horse towed guns around the battlefield, unlimber them and fire grapeshot at infantry or roundball at field fortifications would allow the Mongols to continue their battlefield domination for centuries longer?
Well I suppose if they had the tech, the Mongols could establish manufactories in China to produce the artillery needed for their armies. But that risks Chinese rebels getting their hands on it as well...
They'd probably use horse artillery for scaring the enemy and driving them away from strategic points, rather than bringing them in close and firing which would require constant drill and training to do right, which would require great leaps in organizational structure which the Mongols (nor anybody else) probably didn't see the need for. Mongol horse artillery would likely be noisy, smoky and used in tandem with rockets, following on from the way Chinese militaries used gunpowder.
But I doubt it would enhance the Mongol ability too much. What they historically conquered are pretty much the areas where horse armies reigned supreme - an invasion of Burma is still going to be troublesome for the Mongols because of the climate, horse artillery or no. Pushing further into Europe would require heavier cannon than horse artillery to deal with the various fortified settlements. I suppose they could have made gains in Egypt, but then again Ain Jalut was more the Mongols falling into a Mamluk trap rather than any technological deficiency in their part.
As for maintaining their dominance... well certainly horse artillery would have enhanced their capabilities,
if they could continue to make/replace them. Course by the time of the 14th C Mongol rule was collapsing everywhere, and they simply didn't have the production capabilities to make enough cannons to resist more sedentary countries.