An interesting question! Difficult to answer, because our examples are few and often affected by strange circumstances. Below is a chart of cross-cultural marriages among the elves (let me know if I forgot one)
(They’re color-coded by culture: red=Noldor, purple=Vanyar, blue=Teleri/Sindar, grey=Avari, green=multicultural)
As you can see, in most of these relationships the wife goes to live with her husband’s people. However, this is not always the case (Finarfin famously lived among the Teleri, and Galadriel and Celeborn seemed to take turns moving to different places for the other.) The problem is that most of these relationships are affected by special circumstances - mainly, that many of the elves above were in positions of power that would have necessitated their living in a certain place. Finwe couldn’t very well go live among the Vanyar, being High King of the Noldor. Elrond was lord of Rivendell. Turgon was committed to following his father into exile, so Elenwe had to choose either to follow him or stay in Valinor. And Aredhel was held captive by her husband, so that’s hardly a fair example.
Unfortunately, Tolkien wrote about so few “normal” elves that it’s really hard to tell - the majority of his characters were members of the royal family, and so politics had a huge impact on their personal lives. Even so, I think the example of Finarfin and Earwen prove at least that it wasn’t always the wife who went to live with her husband’s people.
SOURCES: The Silmarillion, The Histories of Middle Earth vol. 12 (“The Shibboleth of Feanor”)