These migrations occurred as a result of political struggles affecting the nomad groups.
For instance, the Alans were driven out of current-day southern Russia and into central Europe by the fiercer Huns.
The Huns, in turn, had gotten used to the fruits of "civilization" in the form of tribute from Roman rulers. When they tried to collect similar tribute from the Persians, the latter resisted successfully, even counterattaccking them in southern Russia. This led them to greener pastures central (non Roman) Europe, until the Romans combined with groups like the Franks to defeat them.
Similarly, the Magyars migrated to central Europe early in the Dark Ages after an unsuccessful rebellion against their (Bulgar) overlords in southern Russia. Finally, the Mongols moved west after Genghis Khan united them in a search for a better life than the steppes could provide.