Reddit reviews Training the Roman Cavalry: From Arrian's Ars Tactica (Military Series)

First off, I'm still looking for more sources to read on this myself so my recommendations are merely places to start rather than a comprehensive list.

One of the better history books I have enjoyed are the books by Ann Hyland. She combines a love of history with her love of horses and approaches things from a perspective that, even 4000 years ago, horsemen were mostly practical people (just like today). The place you may want to start is The Horse In the Ancient World as that will cover the Mesopotamia, Egypt, and Anatolia regions. She has a completely separate book on Rome (2 actually, Equus: The Horse in the Roman World and Training the Roman Cavalry: From Arrian's Ars Tacticta).

Compared to other history books on cavalry and warfare, I find Ann Hyland to understand the mindset of horses and horse people better than traditional college professors and historians. Although historians can gleam the how and what from their sources, I find they don't necessarily understand the why of horsemanship practices as they don't usually have personal experiences around horses.

Another, slightly more general, book on warhorses that I have read is by military historian Louis Dimarco. He wrote War Horse: A History of the Military Horse and Rider. He goes through history covering noteworthy periods from the rise of chariots, the Mongolian horde, the American Indian wars, to even chapter on the use of horses by American forces in Afghanistan. His book will focus far more on military applications, whereas Hyland will focus a bit more on the equine development.

Both of these authors have good standalone books, but to really understand the topic, I highly recommend you read both.

Arrians Ars Tactica references Alan horses as being superior, hardier species, and describe steppes ponies. Ars Tactica describes a lot of similarity between Alan and Scythian horsemanship. I think its fair to assume that the Scythians similarly had ponies bred for the steppes, resembling Mongolian horses.

Related trivia- apparently words for rider, horsecloth, stirrups, bridle, saddle girth originate from Iranian. The Scythians were thought to be descended of Iranian origin, spreading north of the black sea and then westward, Romania was described as Scythia Minor.

Source: Training Roman Cavalry- From Arrian's Ars Tactica
and
The Horse in Human History