There were plenty of roving groups of Indo-Europeans in the Bronze Age. The Hittites seem to have taken over a non Indo-European group. Although the Mitanni spoke a non-IE language, it's clear their ruling class was Indo-Aryan (their names are Indic, and their gods are recognizable as being from the Rig-Veda). The Medes and Persians settled in modern day iran as well.
The bottom line is virtually anyone could have done it, particularly because the Indo-Europeans seemed to have military advantages, such as the Chariot, that many other groups didn't have during this period. The biggest issue would be twofold. First cracking the nut of the Fertile Crescent. Secondly, coming up with some rationale for moving past this. It's not as if Arabia was a land which was much worth conquering, particularly for horse-based pastoralists.