I doubt that horse milk was the major source of calories, since horses only produce small amounts of milk and the foals need to be around too. Fermented horse milk (Airag) is a (mildly alcoholic and relatively expensive) specialty beverage in Mongolia today, but certainly not a staple food.
I think it is more likely that they brought dehydrated meat and dairy with them, which are common in Mongolia to this day.
Probably they also added to their diet what they found along their way, at least meat. Maybe they even tried some grain.
Milk would include other dairy than Airag, such as camel, sheep, yak and processed durable dairy, such as butter fat, dried curdles and cheeses. These store for long periods of time. Dairy in many forms indeed is a major source of calories in the traditional Mongolian diet that some rural people eat still today.
As far as I can remember, Airag is only produced during July and August. That would be pretty limiting. It also spoils pretty quickly. After a while in warm weather the smell is unbearable. People save the microbes for the the next run in the following year.
I think it is more likely that they brought dehydrated meat and dairy with them, which are common in Mongolia to this day. Probably they also added to their diet what they found along their way, at least meat. Maybe they even tried some grain.