In the 100s BC, Germanic tribes were pushing in to Gaul enough that it gave Rome pretext for invasion. Germanic tribes also went north, supplanting the natives in Scandinavia.
In the 4th and 5th century, Goths and Lombards and Vandals moved West, leading to the fall of the Western Roman Empire. The 5th century also saw the push by the Huns (perhaps causing the former?).
Around the 7th century, the Avars, the 12th century the Mongols.
Not to mention the continual invasions in to Anatolia of steppe tribes/peoples.
Was the birthrate lower in the Roman Republic/Empire? Is the Steppe particularly fertile? Am I basing this question off of an incomplete understanding?
This is a good question with no possible good answer. In each case, the variables were different, and we should add to this the hypothesized "migration" of the Proto-Indo-Europeans into Europe, from some homeland variously located in the western Russia forested steppe or perhaps from Anatolia. Nobody really knows exactly how it happened, but the linguistic data demands a "homeland" of some sort. If that is the case, and just for simplicity we say for now that it was the western Russian steppe, then we know they did not just go to the West: the went south, into Iran, and East into India and even the deserts of western China. They had really scary new weapons at their disposal: the wheel, the horse, the metal plow, cruel social stratification, and the knowledge of how to weave wool into cloth.
You mentioned the Germanic invasions of the late 2nd century BCE. No ancient author seems to know why exactly the Cimbri "migrated," or from where, and we can these days safely put aside Poseidonius' assertion that they left Jutland because the sea was flooding in. Overpopulation, as you guess, could have been a contributor, but there isn't enough evidence to say for sure. Caesar gives us a relatively thorough account of another such migration, of the Gallic Helvetii in 58 BCE, and even then we do not really get a clear understanding of why exactly they decide to pack up their entire civilization and risk the ire of their neighbors, Roman and Gallic, to seek a new land. Caesar reports that they got sick and tired of being harassed by the Germans to their east.
Harassment was probably the reason for the mass confusion of Eastern Europe at the end of antiquity. As you guess, I think most scholars uneasily accept the Huns to be the bogeyman which sent everything in their path fleeing westwards. Post antique movements into Europe had their own inputs. The point is: there is no one unifying impetus for it all. The viewpoint which makes it seem uncanny springs from a Eurocentric point of view (all these pushy others appearing from the lands beyond rising sun). If we shift the view to, say the Volgue, or the Kyrgistan, we get a very different idea of the same events.