"Race" might not be the correct word here, but I'm not sure that ethnicity is the right word either, and I can't use nationality because Normandy is not, nor has it ever been a nation.
I may just be ignorant in assuming that people no longer identify themselves as Norman, but I have never heard anyone call themselves as such. It seems that people from mainland Normandy are French, and Anglo-Normans identify as British. It also seems that the Norman language has all but died out, and with it I would imagine much of the remaining culture.
Is this solely to blame on the hundred years war? Is it that people did not want to consider themselves Norman because of taboo? Or does it simply boil down to the fact that a current day "Norman" wouldn't have a country to call home?
The Normans come from Viking descent and settled the area now known as Normandy, they were legitimized as land holding vassals, Rolo to be specific, to the French Crown in about 917.
From that point on it appears that the Normans spread out from Normandy with a handful heading south to Italy and then north in 1066, probably the more famous conquest.
The Conqueror fought with his eldest son for Normandy before his death. Upon his death, his continental holdings and England were split between his sons. Robert got Normandy, William got England, and Henry got a fat stack of cash. A lot of things happen but the basic gist of it is that Robert miss managed the duchy and sold out to William Rufus to go on Crusade. While he was gone, William died and Henry claimed the England and was prepared for Roberts return, eventually capturing Robert in battle.
Henry I was widely successful but failed to produce heirs after a bit of a tragedy and when he dies, England and Normandy go into a kind of civil war until Henry II, his mom, dad, and some uncles are able to get him onto the throne against Stephen (Henry I's nephew), but at this point the Normans become the Plantegenants.
The Normans in the south, primarily Sicily, are a big part of Italian, German, and Byzantine politics until the destruction of the house around 1194.
From what I understand, Norman language variances died out shortly after they became land holding legitimate vassals. I don't have the knowledge to comment on why.
The 100 Years War is significantly later than the Norman proliferation in France.
Not flaired, not a historian, and sorry if my sources aren't in the correct, but I am referencing:
The Capetians in France: Kings of France 987-1328 Bradbury The Normans in the South 1016-1130 & The Kindom of the Sun 1131-1194 Norwich (should be noted he isn't a historian though) Henry I, Hollister Henry II, Warren King Stephen, King William the Conqueror, Douglas
The Normans always tend to merge themselves into the cultures they conquered, adopting features of the conquered as seemed most practical. It only took a few generations for the original Norse/Danish culture of the Normans settling in northern France to become very much secondary to a version of Frenchness.
Then, in England, William I's Normans adopted features such as governmental practices from the subdued Anglo-Saxons. Similarly, in Sicily and Italy the Hauteville Normans adopted a multitude of official practices that already existed there, and some historians hold that they even took up speaking vernacular Arabic as it was the most commonly spoken language in Sicily at the time. Certainly they administrated in Greek and Arabic (Sicily's pre-existing languages) as well as in the more conventionally 'Norman' Latin. Mark Hagger argues that by the time we get to the second generation, the Italian Normans consider themselves 'sons of Normans' rather than actually Normans themselves.
Anecdotally, there's logic to that. Born and raised in England, parents born and raised in England, grandparents born and raised in England - the fact that there's Irish, Welsh and Italian back there somewhere doesn't mean I consider myself any of those things.
My point is that right from the outset the Normans made a point of merging with those they conquered, at least to an extent, and in that light it's not particularly surprising that the Normans of England and Italy began to see themselves as English or Italian as successive generations continued to be born there and never set foot in Normandy.
As to the Normans of Normandy itself, I can only offer conjecture. As a part of France, Normandy's people gradually came to emphasise their French national identity over their Norman regional identity. This, as I say, is conjecture. Normandy itself is only tangential to my main interests.
Sources:
Angold, Byzantium.
Breese, 'The Persistence of Scandinavian Connections in Normandy in the Tenth and Early Eleventh Centuries', Viator 8.
Hagger, 'The Earliest Norman Writs Revisited', Historical Research 82:218.
Hagger, 'Kinship and identity in eleventh-century Normandy: the case of Hugh de Grandmesnil, c. 1040–1098', Journal of Medieval History 32.
Matthew, The Norman Kingdom of Sicily.
Johns, 'The Language of Islamic Art', Oxford Today (don't have vol./issue/date handy).
Johns, Arabic Administration in Norman Sicily: The Royal Diwan.
Oldfield, City and Community in Norman Italy.