I would either say "Mongols are born on, grow up on, and die on the back of a horse", or "Mongols are born, grow up, and die on the back of a horse"While "horseback" is used correctly, you should use an article such as "a" with "horse."
"Mongols are born, grow up on, and die on the back of a horse" or "their horses" would sound appropriately epigrammatic to me.
Oops! I'd forgotten about that particular aspect of Mongolianness (I don't come across many Mongolians in my daily routine).they often ride without a saddle
"Mongols are born on a horse, grow up on a horse, and die on the back of a horse."
Can't you just hear the hoofbeats? Feel the rocking movement of the horse galloping?
For the rhythm:I'm horse-sick, Nunty, horse-sick.
You said on the back of a horse as well!