I was wondering the same things : I often heard that the relative civilizational collapse that happened in europe during the middle ages was correlated with lower overall temperatures, e.g. what you see here [1] or there [2]. In the second one, the most recent projections (in red), show that there were a lot of very cold years (down to -0.6 degrees relative to normal) during those times. Later on, starting in the 1200s, there was the Little ice age, where temperatures fell to -0.9 relative to normal (I don't know how they defined normal). The peak, when almost all years were at -0.6 or below on that scale, caused the "General Crisis of the seventeenth century" where people from the Goth/scandinavian regions also had huge famines [3].

So my guess would be that there were several very cold years in a row, which caused famines, and the populations understood that their best survival chance was to migrate south, where they had to compete with the roman empire for land.

[1] https://fr.wikipedia.org/wiki/Fichier:2000_Year_Temperature_... [2] https://fr.wikipedia.org/wiki/Fichier:2000_Year_Temperature_... [3] https://en.wikipedia.org/wiki/Great_Famine_of_1695%E2%80%931...