Use of mounted infantry is a viable tactic for sedentary societies and nomads alike. There were some cases where an army was majority cavalry - such as Black Army of Hungary, Byzantine Tagma and a couple of others. In other cases, army was cavalry with mounted infantry - I believe Belisarius had used that combination, and English armies of 100 years war were often entirely mounted, but longbowmen and such would dismount before battle. Combination could also be used in impromptu situations; Byzantine Emperor Basil II mounted infantry on mules when he had to relieve Aleppo in hurry. Journey which should have taken 60 days he completed in quarter that time, but out of 40 000 men, only 17 000 made it. It should be noted that this was travel through very mountainous terrain. You can find more on logistics here:
I do not think however that nomads would have mounted infantry by default. Nomads had very sophisticated tactics, which included heavy cavalry, light lancer cavalry and light missile cavalry (mounted archers). These were all highly trained, and could be used as infantry if situation required it. Sedentary societies also sometimes pulled that off: Byzantine kataphraktoi were trained to fight as infantry if necessary (at least those of Belisarius were), English men-at-arms also occasionally fought as infantry, and so did Polish Winged Hussars. As such, having strictly mounted infantry is not really necessary if you already have cavalry. But more to the point, in open, flat terrain, cavalry reigns supreme. I had spent some time researching infantry-vs-cavalry clashes for my blog, but long story short:
1) heavy cavalry beats all infantry
2) missile cavalry beats heavy cavalry
3) missile infantry beats missile cavalry
Main reason why pike squares could stand against heavy cavalry was the general lack of discipline and training of the latter, what with them being composed of nobles. But well-disciplined heavy cavalry - such as French Compagnies d'Ordonnance and, later, Polish Winged Hussars - was capable of disciplined mass charges. As such it could, and did, break infantry pike squares, especially if you combined heavy cavalry with missile troops (e.g. horse archers). Disadvantage of cavalry was that its effectiveness is highly dependant on terrain. As I already stated, cavalry reigns supreme in open plains. But send them into mountains, canyons, forrests, riverbeds? They lose their advantage. That is what happened to Mongols: they beat Hungarian-Croatian army stupid at open plains, but they achieved absolutely nothing once they came to rather mountainous Southern Croatia.
This, of course, assumes that nomads have stirrups - which, if they had managed to build an empire, they probably do. Cavalry without stirrups is rather less effective in fighting from horseback, either with melee or missile weapons, though high-backed saddles, or four-horned saddles, do help somewhat. It should be noted that it was precisely nomads - Sarmatians - who introduced stirrups to Romans. Before that, cavalry usually dismounted to fight, or else fought with missile weapons (e.g. cavalry at Hastings used throwing spears, I think). Melee combat was not exactly out of the question, but couched lance technique was straight out - "charge" of pre-stirrup heavy cavalry consisted of trotting up to the enemy, slamming into them with their horses, and then using spear for stabbing, or else other melee weapons. For example, Byzantine kataphraktoi - which may or may not have stirrups - were shock cavalry... which fought with maces. Cataphracts with lances were positioned on flanks and used to guard from enemy cavalry, not to engage enemy infantry. Attack on infantry was done by a mix of mace-wielding melee cataphracts, and bow-wielding cataphracts behind them. If you want details, you can read Eric McGeer - Sowing Dragon's Teeth. Byzantines often fought nomads, and it is not out of question to think that some of their cavalry tactics were adapted from nomads. Further, The Byzantine Art of War, from the same author, gives details on tactics and organization of some of Byzantium's enemies, as well as Byzantine responses to the same. This also includes nomadic groups, though he does not go into detail of particular groups.