Did the Keltoi migrate into France or were they people already living in the area that adopted Celtic culture?
As far as we can tell, there wasn't radical changes in population from the Chalcolitic to the First Iron Age, including along the mediterranean coast between Pyrenees and Alps. Both archeogenetics and archeologic material seem to support the idea of continuity along the Bronze Age and the First Iron Age, i.e. the period we don't have historical information about.
The general consensus is that at least populations partaking in the Urnfield archeological horizon; which in southern Gaul pretty much derives from Malhacien and later local cultures; but quite possible from the Atlantic Bronze Age as well with the idea that proto-Celtic populations might have emerged from the Bell Beaker horizon, and that they were either proto-Celtic or para-Celtic (which is a pretty term for speaking something closely related to Celtic, but not Celtic, we don't know much about).
Now, when the Greek authors first used the term Keltoi (maybe out of an hypothetic Celtic *Keltas), another term was previously used for local dwellers : Ligyes or Ligures. But this name isn't reflected by archeologic or even clear linguistic differences : the consensus there as well support the idea "Ligyes" named an heteroclit set of people that while not sharing a same language (likely comprising Iberic, Celtic, Italic speakers, maybe with some other groups) butdefined by taking part in exchanges with Etruscans and especially Greeks. "Ligurians" would be thus the Barbarians of the shore where Greeks interacted with.
What does it make of Keltoi, as they appeared "only" in the Vth century BCE? Giving the lack of evidence for large migrations, again both archeologically and genetically, it is more and more suspected that they're an indigenous formation, possibly as a coalition of peoples trying to establish some sort of common front in their dealing with Greek merchants and settlements in Provence (not unlike it might have been the case with Elisykoi in Languedoc) : indeed the trade in Provence fueled a lot of connection deeper north where the influx of Mediterranean products is made clear in the various chariot burials (Vix or, more recently discovered, Lavau) but as well the societal development evidenced by the constitution of proto-agglomerations (Aravicon, Hunerburg, etc.) connected with the Gaulish meditteranean arc; making control and accessibility to this trade a clear political interest.
It doesn't mean that there weren't cultural disrepencies between Mediterranean peoples of Celtica, but their relevance to a "Celtic" identity isn't necessarily well relevant. It had been proposed that this "Celtic" coalition became successful, prestigious and attractive enough it ended covering a fair part of the region we call Gaul before the migrations in Northern Gaul of the IVth and IIIrd centuries. As such, it's not that "Celts" expanded over western Europe during Hallstatian times (a perspective which is essentially obsolete academically), but that the Halstattian and LaTenian features from Central Europe expanded along already present connections (which is quite different from what happened in Eastern Europe) and Alpine roads; with a first "Celtic" identity emerging in southern Gaul by the VIth/Vth centuries BCE.