It is a long way from Kazakhstan to Kentucky, but the journey to the Derby may have started among a pastoral people on the Kazakh steppes who appear to have been the first to domesticate, bridle and perhaps ride horses — around 3500 B.C., a millennium earlier than previously thought.
Archaeologists say the discovery may revise thinking about the development of some preagricultural Eurasian societies and put an earlier date to their dispersal into Europe and elsewhere. These migrations are believed to have been associated with horse domestication and the spread of Indo-European languages….
See also this as Scientific American
I seem to recall from J. P. Mallory’s In Search of the Indo-Europeans that the Sredny Stog culture (c. 4,500-3,500 B.C.E.) showed some evidence of horse domestication, e.g. at Dereivka:
I also vaguely remember (in the same book) some ambiguous (and disputed) claims putting horse domestication back to 6000 BCE, although I can’t call the details to mind at the moment.