Paul W. answered 04/04/19
Dedicated to Achieving Student Success in History, Government, Culture
The answer to your question is complicated by the scope of your inquiry and by some of the assumptions contained within your question. The nomadic tribes of the Eurasian Steppes were, as with virtually all peoples, a product of their environment. The great stretches of grassy plains were best suited to the raising of herds of animals, rather than the growing of crops, the animals relying on grazing for their sustenance. This was the origin of the nomadic lifestyle of these tribes; the grass in a given location was quickly exhausted, requiring these tribes to move in search of new pastures. The need to repeatedly relocate meant that these tribes limited to their possessions to those that could be easily taken along.
Following the domestication of the horse, the culture of the tribes of the Eurasian Steppes was transformed into one that was centered around this animal, which provided them with both food (dairy products from the milk of mares) and mobility. These tribal people practically lived in the saddle, spending more time on horseback than on foot. As with all things, the more experience one gains with working with a horse - learning how to communicate with one's partner so as to be able to control his speed and direction - the better one will become as a rider and, given the nature of their lifestyles, the tribes of the Eurasian Steppes produced the most accomplished riders in history. At some point these tribes developed a highly effective bow - the compound bow - that could be fired while on horseback. Their use of this weapon, in both the almost constant warfare among rival tribes and in the mass hunts in which they engaged (to provide a supplement to their diet), led them to become expert mounted archers.
This, then, was the secret of their success. As nomads, these tribes could move at a moments notice, traveling considerable distances. Thus, they had what we would call strategic and operational mobility, mobility that was generally superior to the infantry based armies of most European civilizations. On the tactical level, that is, when opposing forces actually engage in a battle, the nomadic tribal forces also had an advantage in mobility. Relying on their powerful compound bows, swarms of mounted archers would charge the enemy formations and, once they were within range, shower the enemy with a barrage of arrows. If the enemy attempted to launch a counterattack, the mounted archers would simply ride away to a safe distance, usually faster than the enemy could follow them. Small wonder that European armies found the nomadic tribal peoples difficult to defeat (nor was this only the case for Europeans. The peoples of the Middle East and of China faced the same difficulties).
However, the nomadic tribes of the Eurasian Steppes were far from invincible and faced their own limitations that curtailed their success. It's no accident that the expansion, for example, of the Empire of the Mongols ended on the edge of regions with great deserts, forest, jungles, and mountains. The great herds that nomads like the Mongols depended on for their sustenance themselves, in turn, depended on access to vast amounts of grass for food. Nomadic tribal forces could launch a temporary invasion of locations in Europe, but they couldn't occupy portions of Europe without having to abandon their herd dependent nomadic lifestyles. This is why such peoples as the Huns, the Magyars, the Mongols could cause havoc in Europe, but could not dominate Europe for any length of time. The Magyars are an example of a nomadic tribal people from the Eurasian Steppes who actually settled in Europe (origins of today's Hungarians). Significantly, the Magyars abandoned their nomadic, exclusively herd based lifestyle for a settled way of life dependent on growing crops.
Another weakness of the nomadic tribes was their inability to unite under a single leader for any length of time. It's a tribute to the leadership abilities of men such as Attila and Genghis Khan, but it's no accident that such figures are few and far between in history. The nomadic tribes of the Eurasian Steppes excelled in warfare, but it was constant warfare among themselves that kept these peoples divided (indeed, one of the chief strategies employed by the Chinese against the nomadic tribes was to encourage conflict among them).
Lastly, you asked why the peoples of Europe weren't pushing East. Your list - Celts, Ostrogoths, Teutonic Knights, and Slavs - includes very different cultures with very different motives and means. Simply put, peoples from Europe did indeed push eastwards. Among those you named, however, it was the Slavs of Russia who ultimately came to dominate the Eurasian Steppes. This was a centuries long process and came about only after the Russians freed themselves from the domination of the Mongols (who ruled the Russians through the cooperation of Russian leaders).