I think it depends on what you mean by "Mongols," the Mongol Horde as pertaining to the one that Genghis had I don't think used any sort of ground military at all, they were essentially the blitzkrieg of their time; thus they didn't need an infantry. Any fortified cities they had to take over they would starve out by destroying and eating the near by villages up.

After the Mongol empire was established (from what I know; which is just me having poking around the mongolian history for about 5 years now) that they used any and all armies via psychological warfare and just natural inclusion. The mongol mobility was why they were a strong and powerful force, being able to get to where they wanted to attack; scavenge and build siege weapons (so they didn't have to bring them along,) were the main assets to the mongolian destructive force. Before any of the armies could build up a force big enough and get to the place they wanted to protect it would've been starved to death or destroyed pillaged by the time they got there.

I know for a fact that every Mongol warrior had more than one horse, all of which could've been used for military purpose (which is different than the european horses; having one for combat the other for travel ect ect.) And thus I don't believe you would've seen very many cavalry being dismounted as they would just get on a different horse. The mongols fought differently than other armies at the time because they could; a blend of the way their command system was, the tactics they used, and the mobility is why they were superior during their reign.

So as for the question, how would the foot soldiers fight? I think there is a lack of "Mongol" infantry, most of what I've encountered is the infantry of the Mongol army were auxiliaries from other countries like Korea, Armenia, China and ect; which they obtained after creating a large empire. They used kharash (prisoners from nearby towns and past battles) to help take the brunt of the impact of arrows/bolts along with initially breach the walls followed up by infantry when laying siege to a city which they mostly took over using Arab and Chinese engineers.

The 'actual' Mongols didn't really have unmounted weapons which sort of indicates that they didn't emphasise on infantry combat at all, they carried multiple bows with a lot of arrows (sometimes had horses or camels dedicated to carrying arrows) and had a scimitar ideal for fighting on horseback. The heavy horsemen used hooked lances for pulling other cavalry off their horses so they could get trampled and hit with the following ranks of horse archers. As I stated if they were dismounted in combat I think that their number 1 priority would to be getting remounted. However almost every other army they would've faced would have had more skilled infantry and thus I don't think they would've fared too well.

I apologize if I didn't answer your question well enough. If anyone has any further information I'm quite curious on the topic also.

I am surprised nobody yet mentioned that Mongol Wrestling is one of the Three Manly Sports of Mongolia. Other two being, properly, horseback riding and archery. Mongol wrestling has long and rich history. When it comes to hand-to-hand combat, nothing even comes close to the effectiveness of wrestling/grappling. Mongols, besides revering cavalry and archery, were ardent practitioners of wrestling, even to very royal blood: Ariq Boke's "Boke" is root of the word "Bokh", meaning "wrestler". Continued tradition of wrestling/grappling excellence is exemplified by modern day Mongolians: they have the highest per-capita judo cup points in the world. So when it came to hand-to-hand combat, Mongols were well prepared. 13th century Mongols were forged formidable by constant ruthless warfare within themselves. Superiority of horsemanship cancelled each other when a Mongol faced a Mongol. A Mongol had to be better on all facets of warfare.

As with becoming an equestrian was placed above mastering of his own legs, no doubt a Mongol had to be well versed in bow and arrow early. To talk about the difference between English Long Bows and Mongol Compound Bows, one need to take into consideration who were the archers? On the English side, you have conscripted peasants who never seen any bow before being drafted. On Mongol side, you have a Baatar who literally grew up with the bow, for whom the bow is an extension of himself. Of these two sorts, it is obvious, whose bow would be grossly inaccurate and command lighter draw strength. Would it be of the novice English peasant helpless without an instruction or of Mongol Baatar who grew up on a horseback with the bow?