A poster at Eurogenes linked this from Science Magazine "Ancient DNA upends the horse family tree"
The dominant opinion on horse domestication has been that horse hunters progressively managed herds, earliest in the Botai Culture, and this interaction led to more extensive use of the horse as a work animal for busy hunter-gatherers.
There was already a lot of skepticism to David Anthony's bit-wear analysis ("Botai and the Origins of Horse Domestication" Marsha A. Levine, 1998). Much of the Botai evidence, such as lipid residues or accumulation of horse pies, are interpreted as supporting facts to other facts, whose correct interpretation is supported by those other facts, which contribute to the growing weight of facts that all point toward domestication and/or husbandry at this site. I imagine Heidi Cullen flying over the Botai ranch on a broomstick.
In this paper "Ancient genomes revisit the ancestry of domestic and Przewalski's horse" (Gaunitz et al, 2018), the authors make it clear that Botai and Przewalski's are not in the line of modern domestics (at all). They seem to fall into the same trap of assuming that the Botai horses actually had improved morphological features and that Przewalski's are just 'feral' domestics.
The Backbreeding Blog also questions the notion that Botai horses were substantially improved, being that it was so early and so short of a period that almost nothing about Przewalski's could be described as feral even if they were descended from Botai.
The evidence for horse domestication is probably right under our nose (as seems to be case in Neolithic Europe) and those first horses and asses will be found to be deeply integrated in cattle herding societies that first made use of horses. I'll put money on that.
There is a symbiotic relationship between some members of these animal families that may tell us a little about aurochs behavior and the behavior of Early Neolithic Taurine cattle. Unfortunately, so many of these Eurasian animals are extinct or endangered that it's difficult to find large enough herds in a wild environment to study how the more social animals once behaved. But the above cave painting might be a clue. See this
post Guard donkey.
Gaunitz et al figure about 2.5% of Botai in domestic lineages. That might be a clue to how far west you need to go before you'd see a real horse several thousand years ago. I'm guessing the European Plain.