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Military advantage

Horse-poor populations were at a great military disadvantage - especially in the plains. See Indian example here.

Provenance

Horses were advantageous in the charriot age. Eg. IA invasion.
They increased their advantage in the horse-back age. Eg. kushAna, shaka, huNa, Turkik invasions into India.

Battle tactics

Could rapidly outflank infantry blocks and attack from sides.
Could use archery or long spears to harrass from a distance, while rapidly changing positions, without risking precious horses.

Numbers per rider

Horses can charge full speed about 2-3 miles a day. Fully exhausted after that. So, 2-3 chargers per men needed in battle.
Mongols often used multiple (7-10) horses per rider, and stored meat between saddle and the horse to each on horseback. They’d drink mare’s milk and meat in extreme situations. So all they needed was fodder.
Turks - 2-3 mounts.

Range

Range depended on breed, the weight of the rider (and armor), and many other factors, such as environmental conditions. Mongol ponies, for instance, were bred for stamina - i.e., traveling long hours and distances at a steady pace. The scrappy little Indian-bred horses used by the far-ranging Maratha cavalry in India were similarly bred for long-haul stamina.
Infantry could cover at most 30 km per day.
During the invasion of Hungary in early 1241, they covered up to 100 miles (160 km) per day.
Cavalry in the 19th century, marching and using one horse, would usually limit their daily march to about 40 miles. The troopers also didn’t ride the horses the whole time. They would alternate between riding and walking beside the horses.
There were warhorses that could cover 360 km in 1 day. (“The trade in horses between Khorasan and India in the 13th - 17th centuries” by Ali Bahrani Pour.)

Equalizers

Battle of Agincourt was fought in a place where archers held advantage due to topography. English were helped by topography in most places. Indian northern plains had rarely such topographical advantages.
Guns and cannons made things more equal.