This is a great question! I am most familiar with the situation in Britain during the Roman and Post-Roman period and so will primarily be addressing your question through the lens of the British evidence.

Firstly, I would like to address the idea of Roman citizenship and the exportation of Greek culture to Britain. While it is certainly evident that Roman culture and by extension the adopted Greek elements of this were exported to Britain as it was gradually conquered, for the most part and for much of the population (Particularly in more rural areas), life would have continued much the same as in the Iron age before the Roman conquest.

It is often imagined that once the Romans had conquered a new area that there was a great cultural shift and wholesale adoption of Roman ideals and society. However, this process was much more gradual and involved the coalescing of two cultural groups, which in themselves had much variation. Most academics, particularly in Britain, use the term Romano-British to refer to the population and material culture - because they were exactly this - a hybridisation of Roman culture and British culture. Much of the population continued to speak Brythonic and lived in very similar ways to how they had before the Romans came.

While there were changes to the overarching expressions of culture - forms of Roman pottery are imported and produced locally instead of Iron age wares, Roman types of Brooches appear, Roman towns are founded, Roman coins are used, and Britain is administered as a Roman province - Much of these show influence of local populations through techniques and art styles.

Now, to address the main point of your question. In Britain, the end of Roman control is often attributed to around 410AD with the withdrawal of Roman administration. A main expression of Roman control is the town. Towns in Roman Britain were the apparatus of state - they controlled taxation, trade, and other administrative tasks which supported the Roman state. With the end of Roman administration, the main purpose and function of these places ceases and consequently towns are abandoned. The process of abandonment is one that has been debated for decades - whether towns gradually declined or rapidly imploded - but the result is the same, by 450AD at the latest there is very little detectable activity in towns.

This situation is reflected in the Romano-British Villas dotting the landscape - they are abandoned, and these once economic powerhouses are no longer functioning and by extension no longer supplying goods and produce to towns. In addition, coins are no longer being imported into Britain - typically used to pay wages of the army, for trading, and for taxation. Existing coins begin to be hoarded with the number of coin hoards skyrocketing as people seek to store their wealth until the storm has passed. To complement the end of towns and villas, pottery production seems to end in Britain with very little evidence of continued production in Roman styles or even use of Roman methods - wheel thrown pottery would not re-appear until the ninth century.

These can be seen as key facets of what made Roman-Britain more Roman than British, so with their decline and abandonment, society rapidly changed.

What we essentially have is an entire society grinding to a halt. But how did people react to this? and how do the incoming Germanic migrants play into this?

Well, in the west of Britain - think Cornwall, Devon and Wales - there is a movement towards using hillforts. People begin to occupy old prehistoric hillforts and construct new ones. Cadbury Castle, Cadbury Congressbury and Dinas Powys, are some well excavated prominent examples. At some of these places, at least initially, Romano-British material has been found used as it always had been, although it was no longer being produced, so quickly became scarce. Imported Roman wares from the Eastern Empire were highly sought and are brought into the west of Britain for elite consumption. People continue to value Roman culture but are no longer in a position to create it themselves, and the amount of material we can actually find is by comparison minuscule.

The east of Britain, particularly in England, is quite different due to the appearance of a new group of peoples who bring with them their own culture, objects, language and so on.

New settlements appear using new building forms which are considerably different from what had been common in the Roman period. Historical accounts, primarily from a monk named Gildas as well as the later Venerable Bede, depict a deal gone bad - the Germanic migrants were initially contracted as mercenaries to help defend against raids from Ireland and Scotland, as well as other Germanic tribes. Instead they turn on the Britons and usher in a period of warfare with the Britons losing much territory and getting pushed westward until a final battle at Mount Badon. How much of this is true? well, we're just not sure.

Archaeologically, the Romano-Britons who had inhabited the east of England are almost completely invisible. There has been some suggestion that with the appearance of a new, dominant group, that Romano-Britons become second-class citizens who operate as the lower section of society. Within as little as two generations, the days of Roman towns and mass-produced objects, and the very idea of the Roman state may have been forgotten. Coins for example have been found used as pendants in Anglo-Saxon graves - a departure from their original purpose.

Research into this period has attempted to locate evidence of Romano-British populations through closely examining the workmanship of Anglo-Saxon metal objects to detect whether the craftsman was using techniques found in earlier Romano-British objects - although this is far from conclusive, but does affirm the lack of evidence we have for the Romano-British population in now Anglo-Saxon areas of England. From this lack of evidence we can see that people are not operating in the same ways as they had under the Roman state, with pottery, coins, and other cultural objects falling almost unceremoniously out of use.

To draw this to a close, life for people in Britain didn't continue as it had. It dramatically changed as the foundations of what made Roman-Britain Roman collapsed. People were forced to adapt to new economic conditions with production of crops and goods focused on the much more local level - producing to survive, rather than to supply surpluses to the Roman state. In some places, burial traditions appear to continue much the same as they had been in the late-Roman period although again this only lasts for at most 2-3 generations and is by no means the norm.

Even in areas which do not fall under Anglo-Saxon control, society has rapidly changed in form with local regions controlled from hill-forts and an economy much more focused on the exchange of favours and gifts than the use of money and commerce.

This is a very simplistic overview of an incredibly complex period, with a myriad of different circumstances and consequences playing into a situation which is currently not as well understood as other parts of Roman history. I hope this has at least in part covered your question!

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Sources:

Fleming, R., (2011) Britain After Rome: The Fall and Rise 400-1070. London: The History Press.

Dark, K., (2010) Britain and the End of the Roman Empire, Stroud: The History Press.

Laycock, S., (2008) Britannia - The Failed State: Tribal Conflicts and the End of Roman Britain. Stroud: The History Press.