Initially the horse breed that was the most likely utilized was the Mongolian horse, given that the species has been in residence in the Eurasian Steppe for upwards of 150,000 years. This horse is recognizable short, typically standing in the upper 40 inch to mid 50 inch range.Now I say initially, because as they migrated West. The Huns very well could have changed their preference for the small Mongol horse for something better adapted to the Old World Forests of Eastern Europe.

They are actually short because on the steppe it is evolutionarily advantageous for them to be small The similarly sized Przewalski's horse has never been widely domesticated and occupies much the same range as the Mongol horse. It reduces the amount of food they have to find, it conserves heat and it makes them harder to spot against the horizon.

All of that being said, the horse (as a whole, not just the Mongol horse) kinda evolved alongside the advent of human civilization. The most recent non-horse, horselike relative in the genome broke away around that 150,000-300,000 year mark where the horse and Przewalski's horse separated into different species . In all likelihood we vastly directed the evolution of the horse to benefit our needs as budding agriculturalists.