During the time of the 2 empires, East and West, the surge of Germanic tribes (Visigoths, Ostrogoths, Franks,ect) [and the Huns] constantly invaded the West empire until it's dissolution. From my [limited] understanding this wasn't acquisition of new territory, rather they moved away from their home regions to the Iberian peninsula, Italian Peninsula, and province of Gaul as well as the Balkans.
Further on in the timeline post the collapse of the West, the Lombards moved into Italy and numerous Steppe tribes and Slav tribes also invaded and settled the Balkans.
What caused this relocation of all these tribes, what was going on in the East that made them want to resettle? On a side question where did the Slavic people originate from?
Mainly climate change and the onslaught of the Huns