Pretty crazy that Mongolian armies sustained themselves off of horse milk to remove the need for sustenance supply chains during invasions.
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.
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.
Airag is sold in Mongolia in grocery stores actually. It is harder to find kefir there.
The proportion between lactic acid and alcohol depends on how exactly the kefir was fermented.