To suggest that cavalry moved at the same speed as infantry is more than unreasonable, simply becasue the reverse was true in almost every mass deployment of horse armies, from the ancient times until WW2 (I remember a very long and detailed debate I participated in on the HoI forum, detailing just how much strategic and tactical mobility Soviet Cavalry had compared to other units of the era...they were pretty darn fast over good roads and amazing over broken terrain, which made them pretty decent at escaping encirclements).
Further, it mucks with any country bordering the Steppe - there's a reason why the main invasion routes such as the Crimea-to-Moscow line which went over easy fords and flat grounds was so heavily fortified over time - there was no way to spot and intercept even the very enormous horse armies (100K!) that the Crimean Khan would bring without fixed defenses blocking their way even with Russian armies (which were still fairly mobile and cavalry heavy compared to their western counterparts).
Fighting in Central Asia/Eurasian Steppe with infantry should not be viable until the late 1600s. Reducing horse speed is the wrong way to go about it. If man's vaunted long-distance prowess was a real strategic reality, Horse-Nomad armies wouldn't be as much of a vehicle for building great Empires as they were in history.
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Generally speaking, in Western Europe infantry walked over roads, making them faster, whereas the Horse, even when in significant independent units, couldn't really go around the infantry because of the ditches, hedges, fences and canals that Eastern Europe didn't have; not to mention far more frequent castles and towns.
Even then, small horse units definitely have great mobility. Chevauchee was done on horseback. Russian princes raided each other on horseback. Tang armies (China's most illustrious military dynasty of mostly-native origin) fought on foot but moved around on horse.
Granted, cavalry already has less attrition to deal with, but again, if it's faster than infantry, it's because that's how it was.