Casual ? By all accounts they were pushed away by other steppe peoples. The usual story really - pressure from their neighbors tended to be the easily most common reason nomad peoples started the long trans-Eurasian trek. As for the distance, caravans and travelers could cover the Silk Road in a couple of years. There's no reason why an entire nomad people could not similarly traverse virtually the entire width of the great steppe in just decade or two, especially if they're unable to secure decent pastureland along the way from its current occupiers and thus have to move on fairly soon.

Mind you, I've also read that around the time the Hsiung-Nu left their old haunts, the Huns arrived in European consciousness and the Migrations generally started going, the steppes were suffering from a period of drought which duly drove the nomads to look for new lands.

All steppe peoples were rather similar when you look at the basics - pastoral, horse-riding survivalists with a penchant for mounted archery, raiding and good organization. Permutations of thing like religion, art, equipement, favoured combat technique etc. etc. were then pretty much endless. However, the Huns were definitely new arrival from further east by the time they start appearing in records, and the only larger group known to have started migrating from near China inside suitable timeframe were the Hsiung-Nu. It is also well known that whenever a major steppe people started drifting westwards they usually ended up going all the way to the end of the steppe (and either remaining in the Black Sea region, proceeding to the Great Hungarian Plain, or going south towards Iran, Asia Minor and the Middle East when they ran out of steppe), so it's pretty likely the Hsiung-Nu and the Huns are more or less the same bunch (albeit with all the usual odds, ends and conquered subjects added along the way to the latter).