The "White" Huns (properly called the Hephtalites, AFAIK) were apparently a bit different bunch than the "Black" Huns who came to Europe, but AFAIK they shared territory (and probably to at least some degree leadership) for a while around the time they were approaching the Black Sea region before going their different ways. The terms may just come from the Hephtalites having been a (reasonably) Caucasian people who were part of the Hunnish confederation for a while, and hence "whiter" than the definitely Asiatic Huns proper. Arabs for example seem to have considered most of the steppe peoples "white" (compared to themselves anyway, and certainly in comparision to the Berbers and Africans), so that sort of distinction would make sense. Then again, similar "color schemes" appear in the names of several other nomad tribes that AFAIK had no such circumstances, and the Turkish khanate that rose in the place of the Yuan-Yuan are known as the Gök ("Blue") Turks... so there may not really be much in the way of particular reasoning involved in the names.

But outside the Mongols we aren't too sure if there were that many that made a full scale trek across the steppes.
The Huns, the Bulgars, the Avars, the Khazars, the Oghurs (at least some of whom got absorbed into the Magyars, who then settled in Hungary), the Pechenegs (known also by several other names, about all of them apparently meaning roughly "steppe dweller"), many other Turkic tribes... there was quite enough of them. Whatever dynamic now exactly sent the Huns packing westwards seems to have had enough aftereffects to be causing migrations still at least five hundred or so years later, although I've gotten the impression the situation had more or less stabilized by the end of the first millenium AD - the Mongols were straight empire-building conquerors, not migrating conquerors after all.

My point is, that is is odd that the Huns in a weakened state was able to subjugate every single other steppe tribe on their way to Europe. That makes no sense, and it makes little sense for them to set out to do it after the first victory. Settling in the new lands would be more likely if you have just been kicked out yourself. And they can't possibly have been planning, on the borders of China, to go to Europe. The obstacles were simply too great when you are defeated.
Had they been able to subjugate "every single other steppe tribe" along the way, they wouldn't have built their empire on the Southern Russian steppes - they'd have established it in Central Asia already. It would not, however, by any means have been necessary for them to achieve such feats either - they only needed to win enough and look threatening enough for resident tribes and confederations to allow them to pass through. No doubt if and when they collided into a strong faction and realized the fact (after the initial skirmishes) they mostly tried to work around that area, or negotiate passage; they might also have "clipped" the edges of several such communities, probably taking along or destroying a few of the smaller component tribes on the side.

Mind you, they were probably also well on the path of empire-building long before they got near Europe - but nomad empires have no need per se to remain in one area, and it is entirely conceivable they would have wandered on out of sheer habit if nothing else. They may also have been unhappy with the territory encountered thus far - the eastern steppes at least were suffering from drought and overpopulation (and hence over-grazing of pastures) around the time, and hearing stories of the rich settled lands lying to the far west and south may well have piqued their interest. It's not like the Silk Road was anything new, after all, and even the nomads were no doubt at least vaguely aware there were civilizations far away.

However, it would also seem like migrating nomad tribes tend to just keep going until they find some really rich pickings - a prosperous region to take over, such as the city-states around the Black Sea and the Russian rivers, good pastureland, a weakened empire occupying the above... - or hit an obstacle they can't overcome or circumvent. Oh, a fair few along the way will get killed, remain behind, get absorbed into the local tribes, and all that; but the ones who continue onwards can always refresh their numbers the old-fashioned way and by absorbing small tribes along the road (mind you, the small tribes may be only too happy to sign up into a big, strong group; kinda like joining a gang...).