The idea of a distinct and cohesive French culture, even less nation, didn't exist before the Middle-Ages and even there wasn't an exclusive, holistic and popular identity it eventually became much later. Likewise, we can't draw a clear continuous cultural or linguistic or territorial line between ancient peoples that lived in what later became France.

The broad region was inhabited since the Paleolithic, but before the First Iron Age (i.e. from the VIIth century BCE) we have no historical sources to rely on, our knowledge of the period being essentially sourced trough archeology. The first peoples we're informed about, from Greek sources, were Celts : today, the word took a broader meaning from a linguistic definition (i.e. peoples speaking a set of more or less closely related languages) to labelling a set of historical and folkloric references. But while, among other, Ireland or Scotland are considered quintessential Celtic cultures or nations; "Celt" had a narrower definition, essentially naming the peoples living between Pyrenees and Alps although their kinship with some of their neighbors was acknowledged.

The term which is more used for these peoples are Gaulish or Gauls, from the Roman word for a vaguely-defined group of population living on both sides of the Alps they called Galliae, the Gauls; shortened in "Gaul" nowadays for the region set between Pyrenees, Alps and Rhine before and after the Roman conquest.

Territoriality, Gaul being defined by these geographic borders is already quite different from modern France, as it includes other countries as Belgium, Germany or Switzerland whose peoples wouldn't consider them as such or even that particularly close to France for sharing a same geographic ensemble set 2 500 years ago.

Linguistically, Gauls were a Celtic people, meaning they spoke a language close but indirectly related to Gaelic or Welsh; when French is a Romance language with arguably few Celtic elements (part of it being taken from languages other than Gaulish). It's apparent from both historical and archeological sources that Gauls shared a lot of cultural features with their neighbors, especially in Britain and Germania (i.e. beyond the Rhine) : linguistically of course, but also materially (especially with transrhenan people with who they partook in same material cultures called Hallstattian and LaTenian, i.e. archeological regions defined by a set of produced and consumed products). These relations weren't exclusive to other, especially with Etruscans or Greeks (some of them settling in modern Provence) exchanging prestigious products such as glass or wine in exchange of agricultural (grain, salt, etc.) or mineral products (iron, gold, tin, etc.). Indeed Gaul in Antiquity was one of the most populous and productive regions of the western Mediterranean basin, quite possibly with a population of 10 millions inhabitants by the Ist century BCE.

Unfortunately a great deal of Gaulish culture is lost to us : while they used writing for ostentatious or pragmatical purposes, they seem to have frowned upon using it for their "formal" literature. What we know of their beliefs, folklore, myths either comes from what others had to write about it, what we could take from more or less distantly (both chronologically and geographically) related cultures as Roman Gaul or early medieval Ireland or which interpretation can be made from archeological finds.

Gauls seem to have both thought themselves as belonging to a macro-regional ensemble defined by the aforementioned "natural" borders but also were divided in petty-peoples rivaling each other. This is not particularly uncommon in the ancient world, where ancient Greeks both had a notion of a broader Hellenic horizon, but never shied of fighting each other. Politically, drawing a line between modern French and ancient Gauls is impossible as well : "our ancestors the Gauls", as the old trope goes, is only true in an a posteriori, historiographic sense claiming a national (as in from masses defining themselves as a nation) legitimacy drawing from continuity of settlement (against, notably, a French nobility that claimed descent from Franks). A Frenchman is no more a modern Gaul (or Gauls proto-French) than a Turk is a modern Hittite or a Tunisian a modern Carthaginian.

Gauls themselves didn't as much disappeared than they were more and more deeply integrated within a broad Romanity : even before the Gallic Wars, Gaulish peoples (that weren't already conquered) were under an important Roman influence (up to being sometimes quasi-protectorates). Roman influence was obvious in trade and political influence, with emerging Gaulish nobility being divided in pro or anti Roman factions. After Caesar, and an admittedly brutal conquest where maybe up to one-tenth of the population perished in arms, the emperors followed a policy of appeasement and pacification : both former allies and ennemies in Gaul were gradually given privileges and access to the new order, regional elites being largely maintained and managing it in the new provinces. These were already influenced by a "Roman way-of-life" before, having adopting Roman customs as their ones, and the conquest really accelerated and "justified" it : elites asked and were granted Latin then Roman law to manage themselves, they built villae on the top of their farms, Roman roads were set above the Gaulish roads, etc. In the Ist century BCE, southern Gaul (admittedly conquered decades before most of it) was already considered more of an extension of Roman Italy than any other province.

It's not that locals ceased to be Gauls but what it meant to be Gaul had changed : prestigious access to warfare was made as an auxiliary in the Roman army, drinking wine as a Roman was more prestigious to drink wine as a Barbarian (i.e. drinking it watered down and not pure), with elites Romanizing their cultural expression ceased to be the prerogative of Gaulish-speaking druids and bards and passed to classical rhetoric and deditions in Latin, local gods were still worshiped but in Roman ways, etc. What happened was a creolization, where being Gaul was set on Roman lines : one could show off his druidic ancestry if it was made in Latin; you could remember people chief magistrates were chosen in your family if you requested municipal offices as Romans did, etc.

Maintain of local elites and their connections to lower classes allowed this provincial creolization to flow down relatively efficiently : not much remained of the old Gaulish culture or even language (that might have hold out until the VIth century CE in really remote areas), especially as the new social, political and economical centers of Gauls were not only romanized, but re-created under Roman authority (such as the important agglomeration of Bibracte being abandoned in favor of a now better situated Roman-style town at Augostodonum).

In a way, connection between modern French culture and Roman Gaul is more agreeable on : French is a Romance (or Neo-Latin) language precisely because of these five centuries of provincial creolization and romanization. Modern concepts of state, law, power and order are directly or indirectly coming from conceptions that dominated in the Late Roman Empire. But Romanized Gauls didn't really thought themselves as entirely the same, or even as undifferentiated Romans : there too, identity and political loyalties were bifrons and revolved both on the civitas and on the access to imperial favor or acknowledgement as members of their civitas, of the "Three Gauls" (basically the part conquered by Caesar minus the Rhenan regions) and as Romans.

It's in this context that Barbarians were settled in Roman Gaul : it was already a practice even before the Romans to see transrhenan peoples being sponsored and settled in Gaul, as it happened with the peoples Caesar called "Germani Cisrhenani"; but this was relayed by the imperial settlements of populations either as migrants, refugees, deportees, etc. in the whole of the Roman Empire. For most of the history of Roman Gaul, these didn't seem to have much of a lasting impact, being set in a subordinate position depending directly from imperial good will and scattered more often than not. The crisis of the Third Century saw an augmentation of such settlements in the Empire and in Gaul, partly to make piece for the military difficulties along the Rhenan limes, with groups of peoples banding together, such as Franks.

At this point, none of the peoples emerging on the limes should be seen as homogenous groups set around central features such as language, a single dynasty, a single identity, etc. but rather as more or less opportunistic alliances of more or less related peoples in order to deal militarily and diplomatically with the weakened military and political presence of Rome in the western provinces. Many of these were either living along the limes since centuries, trading, working (including in military service) with Romans; others were attracted by the economic and political opportunities of the borders. These were already at contact with romanity and probably wouldn't have existed without this unequal relationship.