Going to try to give you a simple answer. Well, as simple as I can given competing historiographies.

With regards to Franks to France.

Roman northern Gaul in the 4th to 5th century was already a sort of "frontier zone" rather than a hard and fast "military wall", so there was significant interaction between Romans and barbarians there. It would be hard pressed to draw a firm line in distinction, because the barbarians were frequently recruited into the army to serve on the borders themselves, and because of that would put down roots in the same regions as people who would otherwise be considered Roman citizens. So in northern Gaul (which would later become the heart of Frankish Francia), you had a meshing of military and civilian, which overmeshed with barbarian and Roman.

By the mid 5th century, the Roman state was disintegrating (for a variety of reasons I'll skip) and the barbarians were sliding into their place as functional authority. Sometimes by circumstance, sometimes by force.

This is in fact one of the big questions involved in the Frankish takeover of northern Gaul. Given existing Frankish and barbarian populations in northern Gaul, was the barbarian takeover an invasion, or a civil war, or something else? Because one could argue it was as much a civil war between barbarized Romans vs. Romanized barbarians as it was an invasion of barbarians vs. Romans.

Either way, the Franks followed suit like many other barbarian tribes and took over the Roman leadership slot at the top. After a series of wars in the 6th century, they expanded into burgundy and aquitane, controlling much of what is modern france.

Now one thing to keep in mind, is that this transition from Roman identity to barbarian identity took quite a while, and not at the same pace everywhere. There were still people in the south of France who referred to themselves as "Romans" although we don't know if they thought of themselves as loyal to a lost empire, or they simply thought of it as a geographical designation, like Romagna in Italy. But over the course of 100 years, with no truly strong Roman alternative in western europe (Byzantine Hispania was quite ephemeral and Constantinople was quite far away), people would naturally gravitate in identity towards the closest one nearby that held influence, which would be the franks.

So over time, people just considered themselves Franks, and the land of the Franks became Francia. And whatever ephemeral legal theory the Roman (Byzantine) Empire thought it had over western europe, disappated in the face of harsh reality.

So that takes care of barbarian to frankish transition, but it doesn't cover frankish to french transition. In this case, it has to do with the breakdown of charlemagne's empire.

This part actually has parallels with the shifts in identity during the fall of Rome. So in case you didn't know, the Frankish kingdom continued to expand, and the dynasty in charge went from the Merovingians to the Carolingians, but by the 10th century the Carolingian empire of Charlemagne was breaking down for yet more varieties of reasons I'm going to skip.

By the late 10th century, Francia had split into two halves, west francia and east francia. East Francia would ultimately become Germany, and West Francia would become France. Originally Franks and Francia were viewed as one entity, but similar as with the early medieval transition away from Rome, given the breakdown of a wide scaled state and increasing regionalism, identities (or at least aristocratic ones) coalesced around the two kingdoms.

I'm going to have to apologize if the second half of this is unsatisfactory, I'm running out of steam as my "simple" response has become no such thing, but feel free to ask specific questions.

Some sources:

• Geary, Patrick J. Before France and Germany: The Creation and Transformation of the Merovingian World. New York: Oxford UP, 1988.

• Innes, Matthew. Introduction to Early Medieval Western Europe, 300-900: The Sword, the Plough and the Book. London: Routledge, 2007.

• Wickham, Chris. The Inheritance of Rome: A History of Europe from 400 to 1000. New York: Viking, 2009.

Just to clarify your question a bit, can you tell me which kingdoms in particular you want to know about? And what do you mean by their transition to 'feudal' kingdoms? The term 'feudal' itself is flawed and it is worth checking out these threads for an explanation. They use later examples, but the principle applies here too. There was no monolithic form of government that was universal in Europe, as each post-Roman kingdom had its own history and reasons to adopt forms of rule that are sometimes described as 'feudalism'. Moreover, what is commonly known as 'feudalism' only took hold in the tenth/eleventh century, so are you looking for a political history of Western Europe for this period? Or just more specifically a history of how society functioned? I can help with aspects of this question, up to the seventh century, but such a broad question would be beyond my area of expertise!

Let me know how I can help :)