Horse was a very important game animal in the steppe region from Neolithic times, and it is depicted already in the Khvalynsk culture, from which the Yamnaya culture is derived. Horse domestication followed later - it was necessary for the use of  chariots first evidenced in southern Trans-Urals in the Sintashta culture (2000-1900 BCE), which is very likely to be Proto-Indo-Iranian in language. Horse is likely to have been domesticated earlier than this in the steppe region, and kept for meat, milk and transport (as a pack-animal) but so far there is no clear evidence for its use for riding before c. 2100 in Mesopotamia. As one of the main hunted animals, very swift (āśu : aśva), its name must have been present in the early predecessors of Proto-Indo-European. 

The splendid Majkop culture (c 4000-3000 BCE) of North Caucasus appears to represent a fusion of steppe culture (probably Early PIE in language) and culture of the Caucasus and the south up to Mesopotamia. Caucasus became a crucial source of metal for the steppe area after invaders from the steppe had largely destroyed the farming cultures of  the Balkans and their metal industry c. 4500 BCE. 

Best regards and wishes, Asko

On 21. Aug 2023, at 5.53, Hock, Hans Henrich via INDOLOGY <indology@list.indology.info> wrote:

My initial concern is that the date and time posited for PIE does not square with the evidence of horse domestication and the reconstructed PIE word for horse, reflexes of which are found in all branches. The paper does not even discuss this issue.