I'm going to speak purely about Britain, here, as that's my area.

Late Roman Britain

Britannia, as it was known to the Romans, had never quite become a prospering centre of Roman civilisation, being located on an island far from Rome. However, that does not mean that it did not play host to wealthy 'elites'. The province was divided into thirty administrative zones, known as civitates, with a city, or colonia, at the centre of each. These cities were fairly Romanised, with the typical features you would expect all making an appearance. Camulodunum, for example, had a temple, baths, a sophisticated system of water pipes, and the largest chariot racing track outside of Rome itself.

Importantly, Britannia was a highly militarised province. Through the 3rd Century, a series of fortifications were built along the east coast, and Hadrian's Wall was refortified - thought to be a response to Germanic and Pictish raiding, respectively. A large military force was also generally kept in the province, to the point where Rome began to fear the power that the Governor of Britannia held, and in 210, the province was divided in two.

However, as the Roman Empire's crises grew worse, Britannia too became increasingly unstable. In 287, Carausius, a Roman-Belgae general, seized control of the province, and he and his successor Allectus held it until 296. A similar story played out in 350, with the province retaken in 353. In the 380s, Magnus Maximus attempted to use Britannia as a staging ground for his bid to become Emperor. The general pattern here is clear - Rome was increasingly unable to hold onto Britannia, and by 402, Roman coins stopped arriving in Britannia, suggesting that either the legions had been withdrawn to a man, or that they were no longer being paid.

The final nail in the coffin here came in 406, when Constantine II usurped power in the province. The Romano-British wrote to Rome to ask for help, and were essentially told to deal with it themselves. Rome itself would fall four years later.

The Coming of the Saxons

Unfortunately, we don't know all that much about what happened next. Whereas the Romans had provided historians with a wealth of evidence - large stone structures, dated coinage, many written works, and so on - the evidence for the arrival of the Anglo-Saxons is sparse.

What we do have to go on is a Romano-British writer known as Gildas, who was likely writing sometime during the 6th Century. Gildas was very much a Roman, and was writing somewhere in the West Country - an area that remained under Romano-British control later than the eastern areas of England. Unfortunately, he's not a particularly reliable source, as he was not recording history, but rather, was painting an apocalyptic picture due to his strongly-held religious beliefs

Gildas writes that the Romano-British, under the leader of a king known as Vortigern, were under constant assault from the Picts, and turned to Saxon mercenaries to protect them. The Saxon mercenaries then grew greedy and invaded England for themselves, driving the surviving Romano-British into Wales and the West Country. Bede, a later Anglo-Saxon writer, and more of a historian - he particularly liked giving concrete dates - gives us a very similar account. However, the historical consensus is that Bede used Gildas as a source.

The archaeological evidence, however, does not suggest violent conquest. Rather, we see the Roman colonia mostly abandoned. The cemeteries at Venta Belgarum and Durnovaria, for example, fell out of use by approximately the 420s. There is some evidence that the sites of Lindum Colonia and Durovernum Cantiacorum continued to be used. This is generally regarded as 'squatters' living in the ruins, though they may have constructed wooden structures. Either way, Roman civilisation in Britannia underwent a rapid decline, as the Romano-British were clearly unable to maintain their cities.

Some more rural burial sites suggest that the Romano-British were not driven out by the Anglo-Saxons, but rather, were relatively peacefully inter-settled. At Venta Icenorum, Medbourne, Spong Hill, and several other sites, burials of both Christians (likely Romano-British) and pagans (likely Anglo-Saxons) can be found at the same site.

Eventually, powerful Anglo-Saxon leaders would start establishing themselves as petty kings. In the 620s or 630s, we see coinage being reintroduced to Britannia by Eadbald of Kent. However, it seems fairly clearly that these were newly-powerful Anglo-Saxons, rather than a reassertion of the old Roman elites. For one, many of them claimed descent from pagan deities, such as Woden, even after conversion to Christianity.

Conclusion

Roman Britain had underwent a century of crisis before the Romans finally abandoned the province, and plunged it into historical darkness. It seems that the Romano-British were unable to maintain their cities or field significant fighting forces - the legions likely having been pulled off of the island to defend Rome sometime around 402 - and so the elite lost their power.

We don't know exactly how the new feudal elite arrived or came to power, but it doesn't seem to have been a particularly violent process. Rather, new tribes settled alongside the Romano-British in the ruins of Roman Britannia, and built a new civilisation.

Sources

Gildas - On the Ruin of Britain

Bede - Ecclesiastical History of the English People

Higham & Ryan - The Anglo-Saxon World