[ashark]

That's fine for the first siege (assuming an extra couple months was all you needed), but now you have to supply up with 4x(number of horsemen) for the next one to be in the same position. Tens of thousands of horses for even a smallish Mongol army. And at least some of them must be trained for war, specifically in the Mongol fashion, not just any ol' training. In short, it's not gonna happen on any reasonable time scale.

Plus you lose whatever advantage the extra horses gave you until you get more—I mean, I'm sure they didn't keep them around just for fun. Greater travel speed by resting them off, extra cargo/loot capacity, whatever their reasons were.