This is a complicated answer. If you look at the beginnings of nomadic tribal warfare in the Steppes of Eurasia, you find that Iranic tribes were the original inhibitants of the region. Later on, they mixed with Afasanievo and Andronovo (EHG individuals). These were the two groups that primarily comprised the Scythians and Huns that were overlords of the Steppes and of much of Eurasia. In fact, the Tamga and other such cultural symbols and traditions of the proto-Turks are considered to be directly inherited from these nomadic Scythian and Iranic warring tribes and empires.
Therefore, the primacy of the hegemonic dominance of the region must be assigned to the West Eurasian groups (particularly the Iranian tribes). However, with the passage of time, neighboring Proto-Turkic East Eurasian tribes of Mongolian and Siberian descent for assimilated into their fold, and they became a multi-ethnic confederation with each subsequent generation.
The ethnic composition of the Elites was quite varied, but shows both West and East Eurasian influence, with the West Eurasian influence being heavier. In other words, the West Eurasians were the first conquerors, and the East Eurasians followed suit. The Mongols being the greatest of the East Eurasian invaders, and the Scythians/Indo Iranians/Aryans being the greatest of the Western Eurasian invaders.
However, in the grand scheme of things, the annals of history show that overall, West Eurasians had the largest and the most powerful empires. East Eurasians are a close second.