Gérard Noiriel, a specialist of immigration in France, has a partial answer to this in his Histoire populaire de la France. Although his book starts after the One Hundred Years War, he still points out at the late development of the conception of France as one united country that will only cement itself with the French Revolution. Before that, freedom of circulation in what is today called France would be limited by several powers and by one's provenance and status.
As an Eastern Burgundy villager in 1050, you would have talked another language than French, probably Franc-comtois or Patois Jurassien or even some Germanic language (depending of where in the East you were, most of the East was under the influence of the Holy Roman Empire). Going to the city of Dijon would have been quite a trip, as you would have to travel from your Duchy or County where you lived to the Duchy of Burgundy, in very unstable times for the region. Meanwhile, you would have to cross several "péages" (taxes) and go through the "octroi" (another tax) at the door of Dijon (if you had business in town). If you had nothing to do there, you could have been kicked out as a mere "vagabond".
After having crossed these different borders, you would have noticed that the language was not the same you were used to, as most people would have talked in Bourguignon. You could probably have made your way through the weird language differences (providing you were not Germanic language speaking) in order to speak to them. After all, these languages are all part of the "langue d'Oïl" complex. It would definitely have felt "foreign", but not as "foreign" as meeting someone from Paris (who talked in a weirder language : French). Jews of Dijon would have been seen as a different group, but would have still been less foreigners than you as they had all the rights necessary to stay in town.
As for the "How does a foreigner become "one of ours?"" part of the question. It is hard to tell, but mustn't have been different from moving to a very small place nowadays. You always are a foreigner somehow, but you can get accepted in a tight-knit community. We can see it by the names of people in Medieval writing where the place of provenance is often added to the name. If your name was Pierre, you could have become "Pierre from Dijon" in your new place in the East. Of course, this is provided the Lordship in the fief you chose to move to would have accepted you either as a migrant worker or as a proper serf (providing you weren't already somebody's serf), or villain (providing you pay the necessary taxes).
Thankyou for that information. A few week ago i read a book by Barbara Tuchman. I found that have migratiom streams in France. I was reading a book on the Grail which tell us about migration. Please yell me more.