Do you mean a Government-in-Exile? There's a wikipedia list that I'm too lazy to pull up. Taiwan's also a debatable example, Ming Dynasty took the island from the Dutch in 1662, then political change after political change, and then the Republic of China was ejected from the mainland by the communists. I also seem to remember Portugal's government briefly fleeing to Brazil during the whole business with Napoleon. These are never permanent situations, since people are settled and nationalist sentiments become a factor.
Or do you mean the people/culture moving geographically, not necessarily under the same government? Most examples of this are questionable because these migrations many times come with cultural hybridization and the cultural shifts that inevitably happen over hundreds of years. Are Franks the same as the French, the Visigoths the same as the Castilians/Spanish, Magyars the same as Hungarians? Did Ostrogoths or Lombards have more influence over modern Italian culture? Questions like these start pointless flame wars, especially regarding the Byzantine Empire, because we're talking about lineage here, and lineage is prestige.
Unrelated to the thread, I once knew someone who really prided herself on being part of a Native American people, the Mono. When she underwent one of those genetic tests that are supposed to tell you where your ancestors came from, it turns out that the Iberian peninsula had more influence over her genetics than the North American continent. She was very emotional, denied it and denounced the testers as frauds. She saw it as an insult, but since the Mono homeland was also the northern-most reaches of Spanish colonization of the Americas, it's not impossible that people of Spanish heritage were either forcefully captured or peacefully adopted, and incorporated into the tribe. It's interesting how she defines "part of the tribe" by race/blood, while her Mono ancestors would have only thought of language/customs.