364 Julian's expedition against Persia. He is killed and his successor signs away the eastern half of Roman Mesopotamia and the overlordship of Ibera and Armenia.
372 Eastward expansion of the Ostrogoths creates a reaction from the Huns, who destroy every force sent against them by Ermanarich, the Ostrogoth king. The Huns push forward to the Danube, crushing the Visigoths and enslaving the Gepids. Many Goths apply to the Roman Empire for sanctuary.
376 The Huns push forward to the Danube, crushing the Visigoths and enslaving the Gepids. Emperor Valens permits the Visigoths to cross the Danube into Roman territory to seek sanctuary. They are followed by Ostrogoths.
378The Visigoths revolt against the Romans sheltering them due to food shortages and abuse. While attacking them, the Roman Emperor is outflanked by Ostrogoths and killed with most of his staff. Rome begins recruiting more barbarians to bring their army up to date.
387 Persia returns the Western fifth of Armenia for clear title to the remainder.
395 The Visigoths revolt again under King Alaric and are offered land in Epirus. Meanwhile the Huns launch a concerted attack on the Roman Empire, attacking Thrace, Armenia and Cappadocia.
401 Alaric leads the Visigoths against the Western Roman Empire, but is met by the Roman general Stilicho (a Vandal).
402 Stilicho bundles the Visigoths back into Illyria.
405 Stilicho defeats a coalition of Ostrogoths, Quadi and Asding Vandals descending on Italy from the North, and led by Radagaisus.
406 Entire tribes begin moving westwards, presumably to avoid harrassment by the Huns. In particular, this includes the Quadi and Marcomanni, known collectively as the Suevi, as well as the Asding and Siling Vandals and a clan of Alans from the Caucasus. On the last day of 406 the lead elements cross the Rhine into Gaul.
The New Penguin Atlas of Medieval History, Colin McEvedy, pp 12-13
McEvedy shows the northern part of Pannonia, around Lake Balaton, as Hunnic territory in 406 CE, whereas all of several other maps I looked at show the Roman border intact. Since Attila the Hun was apparently born in Pannonia around 406, I'm led to think that McEvedy was right, but to hedge my bets I overlapped the two territories. The book i'm reading about Attila the Hun indicates that the Huns were allowed to slowly settle that area over the period in question by Aetius.