How many of these invasions were caused by weather?
That's a great question....I don't know the answer though. If the historical assumptions about the Huns and the Mongols are correct, I guess weather (and the lack of grazing land)must have played a major role in their drive toward Europe.
Most interestingly, I am not aware of the historical record of the Huns after they left Rome following the death of Attila.
We know that around 200 AD the climate in the Eurasian land sea turned cold and dry. This sent a tidal wave of barbarian conquest surging against every empire on the Eurasian land mass.
221 AD. Barbarians surge over the Great Wall and overthrow the Han Dynasty.
250ish. Barbarians surge through the Himalayas and overthrow the Gupta Dynasty in India.
350ish. Huns destroy the Ostrogothic state in the Ukraine.
378. Visigoths destroy the East Roman Army and kill Emperor Valens at Adrianople.
400-500 The Persian Empire concludes a century long peace treaty with the Romans so it can focus on fighting for its life against the Yueh Chih in Central Asia.
Etc....