Hey, I am a member of an equine performance group. We have a trio of Marwari horses, a rare and endangered breed of light war horse. I was the resident Marwari rider for several months.
The Rathores of northwestern India were the first to breed the Marwari. Starting in about 1100 or so, they started breeding with a strong emphasis on purity and temperament. They bred out a lot of the prey instinct and the resulting animal was highly intelligent, sensitive, reactive, high-spirited, and highly trainable. They would readily navigate mountain regions, travel through deep sand, attack war elephants, and perform elaborate cavalry moves. Their masters put these horses on a pedestal, absolutely, and held them to the same standard that they held their fellow cavalry warriors. This meant that that there were only three options in how the horses were allowed to depart from a battleground: in death, in victory, or carrying a wounded rider home. There are also stories from antiquity about Marwaris bringing home their men after they got lost in the high desert: Marwaris have exceptional hearing, thanks to their highly scalloped ears, and could detect the sounds of other horses before their riders could. This allowed them to find their way home in a manner that I'm sure seemed supernatural to many of the Rathores.
I can't say anything about horses from Greek or Roman war culture, nor anything from European nations--but the Rathores perfected the breeding of an incredibly trainable and sensitive breed completely capable of developing almost dog-like levels of both education and companionship. Today, our youngest Marwari (who is actually the one whose bloodlines are the most pure and ancient) does lots of liberty performances. In them, he follows his handler in a large and open area--walking when she walks, jogging when she jogs, bowing out when she casts him away with a hand signal, and a number of other movements based on voice and hand cues.
Edit: I just realized that one of our related establishments has a page where they discuss the wildly unusual nature of the Marwari horse, including a bit about its activities on the battlefield. http://www.imh.org/exhibits/online/marwari-horse