My understanding is that while older historiography of the period was a narrative of discontinuity in which invading barbarian foedorati seized control of Western Europe from the increasingly decentralizing Western Roman Empire via elite replacement and replaced Roman culture and institutions with their own, more recent historiography emphasizes a narrative of continuity in which Roman provincial elites invited foederati and hired mercenaries from the frontiers to help defend them after the Empire withdrew its military support, with the Roman elites adopting martial practices both to try to maintain their monopolies on violence and to gain legitimacy as warrior-aristocrats. These elites maintained much of the symbols, culture, etc. of the former Western Roman Empire when such institutions were thought highly of but slowly abandoned them as they lost their legitimacy for more of the trappings of a warrior aristocracy. Archaeology and genetic evidence continues to revise down the size of the so called barbian invasions, which has made wholesale replacement or conquest narratives ever more unlikely compared to those emphasizing transofrmation and accomodation.
In this new narrative of transformation of Western Europe, to what extent did the foederati and hired mercenaries actually become part of the elite as opposed to the Roman elite just eventually turning into warrior-aristocrats culminating much later on in the nobility and royalty of the High and Late Medieval periods? Did many of the secular institutions of the Western Roman Empire used by these elites stick with them as they turned into the medieval nobility?
It really depends of regional situation, giving that Barbarian kingdoms tended to be rather multipolar politically and socially, for instance the set-up of Merovingian aristocracy was importantly changing depending if we're talking of the three main sub-kingdoms, the southern periphery or the autonomous duchies in Germania. Roman nobles in southern Gaul or Spain, for example, could be commanders without necessarily displaying Frankishness or Gothicness (the dukes Mummolus and Desiderius in southern Gaul in the VIth, for example); while militarized displays of northern Gaul Franks aren't necessarily a proof of their military functions but from simple adoption of the codes of the militarized aristocracy in a context of elite networking in the palatial ensemble.
These social transformations weren't really one-way either : the Barbarian militarized aristocracy was importantly romanized, integrated within the late Roman militia before the imperial collapse and probably partly made up of Romans to begin with; while Barbarians could occupy functions more associated with public service and romanity (Arbogast becoming bishop of the city he was formerly comes; Angisbert being senator trough the result of intermarriage between Frankish and Roman families; etc.). Rather than Romans turning into "warrior-aristocrats" (it's fair to say that, at least in peripheral regions, several of them already gained a monopoly on violence trough mercenariate, private levies, etc.) , it's seemingly more of Barbarian elite social codes (naming conventions, clothing, weaponry, etc.) being particularly widespread while being mixed with a late and post-imperial romanitas which gradually made the differences between public service and military elites not that socially distinguishable as long we're talking social groups or families.
Long story short, Barbarian states were Roman states, either directly transitioning from the imperial state or reconstituted from there after their collapse; eventually evolving in a period of social and economic simplification. so the shift of identity might not have been underwent to re-gain an high social status (although there were certainly uncommon case of opportunities of meteoritic social promotion) but to preserve the connection to the state in a period of important social stress and power renegotiation, the proximity with royal palatial network being a main feature.
At least for Francia, this overlapping connection between "frankishness" and romanitas, military and public service, gave birth to a gradually indistinct nobility. As the public service institutions and functions declined in the VIIth century (mostly out of exhaustion as the state was less able to both pay public servants with landed benefits and less able to prevent the appropriation of public land by the aristocracy), Merovingian nobility still held to an identification on Roman legal frames, early Carolingians even attempting to resurrect (and failing to do so) it by associating all of imperial nobility as being part of the militia; which kept this identification (although for display purposes) alive until the Xth century.
This is but one example of course, and the situation could change a lot between post-imperial Britain (where the political collapse was accompanied by a ruined society) or Ostrogothic Italy (which effectively restored the imperial state). But, overall, we'd not be talking of a simply one-way evolution, but where Roman institutions, royal policies, social codes and opportunities bumped into each other into making new societies.