The Yamnaya expansions from the western steppe into Europe and Asia during the Early Bronze Age (similar to 3000 BCE) are believed to have brought with them Indo-European languages and possibly horse husbandry. We analyzed 74 ancient whole-genome sequences from across Inner Asia and Anatolia and show that the Botai people associated with the earliest horse husbandry derived from a hunter-gatherer population deeply diverged from the Yamnaya. Our results also suggest distinct migrations bringing West Eurasian ancestry into South Asia before and after, but not at the time of, Yamnaya culture. We find no evidence of steppe ancestry in Bronze Age Anatolia from when Indo-European languages are attested there. Thus, in contrast to Europe, Early Bronze Age Yamnaya-related migrations had limited direct genetic impact in Asia.