This entire region was Gaul, an integral part of the Roman Empire, at the beginning of the study period. It was a source of slaves, soldiers, merchants, and administrators who could be found throughout the empire and the recipient of similar people from elsewhere in the empire. So, if your ancestors lived here in 500 CE, their ancestors could easily have been from anywhere in the Roman Empire in 1 CE.
Fifth Century Germanic migrations: During the collapse of the Roman Empire, a confusing number of nations used this region as a wrestling mat or passed through en masse. Only two really had a significant effect that we need to account for:
Seventh - Ninth Centuries - Kingdom of the Franks: Starting from their base in northern France, Bavaria, Franconia, and the Rhine valley, the Franks created an expansionist kingdom whose zenith under Charlemagne (768-814) encompassed all of modern France, the low countries, western Germany, Austria, and northern Italy. If your ancestors were living anywhere within this kingdom in CE 1000, then their ancestors could have been anywhere in it in CE 500. The Kingdom of the Franks could have ushered in a general recovery of Western culture but for one problem...
Tenth Century - the Viking Age: During the late 9th and 10th centuries, this region felt the full force of Scandinavian raids. These ended in 911 when the Norse leader Rolf (AKA Rollo, Hrolf, etc.) landed with a large army and was given what became the Duchy of Normandy in northwestern France in exchange for the promise of no further Scandinavian raids on France. The Norse who settled there became the Normans - culturally French people of Norwegian and Danish ancestry. If your ancestors lived in Normandy in 1000 CE, then their ancestors probably also lived in Norway, Denmark, and southern Sweden in 500 CE.
From this point onward, there have been occasional incursions into this region by foreign armies (English, German) but without lasting impact on the population.