Because of my career, I've always been very interested in domestication of animals and particularly horses. I do appreciate the emotional elements of this discussion, and don't want to hijack that element, but thought I would talk about horse domestication.
During the Pleistocene (over 10,000 years ago) in Europe (and North America), horses were undoubtedly food resources that were actively hunted. Current research indicates horses were first domesticated on the Eurasian Steppe (probably the Botai culture of northern Kazakhstan), although molecular evidence indicates that there were multiple events of domestication (i.e., one group didn't do it first and all other domestic horses come from that one group). Another factor that confuses this is that prehistoric breeders (just like more modern breeders) probably used wild stock for breeding with domestic animals.
Horses were the last of the common Old World livestock to be domesticated (sheep, cows, pigs, etc.)
Work in 1988 (Forsten) suggests all modern domestic horses originate from Equus ferus a small statured European equid. Przewalski horses (E. przewalski) have a different chromosome count (2n=66) compared to domestic horses (2n=64), meaning they are not the progeneter of modern domestic horses.
Some authors have theorized that, in fact, nomadic tribes captured wild individuals tamed and broke them, without ever moving into full domestication. The advantage being the individual doesn't have to do much husbandry except for the working horse (no foals, no pregnant mares, etc.). Regardless, initial domestication probably focused on using the horses as primarily food, with transportation forming a secondary use.
Archaeological evidence shows an increase in horse remains from Eurasian sites at about 6,000 Years before present (YBP) that may be linked to initial domestication. Dental wear evidence from these populations may indicate some of these horses were ridden (6,000 years ago). [Interestingly, donkeys were domesticated well after horses, with some evidence from the Sudan at 5,000 YBP and good evidence from Syria at 4,300 YBP].
Domestication of horses, initially for meat, milk and leather, may have required use of horses for their husbandry, but it also resulted in major changes to the societies. Being able to move over more territory, faster, and to carry supplies, led to expansion of cultures. Horse domestication has been argued to have played a key role in the expansion of Indo-European languages (the ancestors to English, German, Greek, etc.) from the Iberian Peninsula to India (although other authors maintain domestic crops were the impetus for the language migration).
There are over 170 horse and pony breeds today. Researchers suggest that stallions were moved and traded, but the "native" broodmares were the foundation of Eurpoean and Asian horse breeding. If true, this would assist in the genetic identification of the origin of horse breeds; however, the record actually reflects that extensive translocation of horses from one region to another have occurred. Viking horses (1,000 to 2,000 years old) from around the Baltic Sea indicate that at that time genetic variability was very large within domestic horses.
Within breeds, there are microsatellite genetic markers that allow distinction of the breeds with a high degree of accuracy. However, this appears to be related to paternal DNA (the stallion) and breeding within a breed, rather than maternal elements.
As an interesting statistic, there are approximately 300,000 thoroughbreds worldwide, but genetic evidence suggests 95% are from a single founding stallion, and 10 founding mares account for 72% of maternal lineages.