This graph from Pew shows the religion of one's partner or spouse. Ethnocentrism is often discussed on this blog, but the religious aspect of it is usually left out. I've presented evidence in other posts that the following groups have above-average levels of ethnocentrism: blacks, Mexican Americans, Jews, Chinese Americans, and Asian Indians. In addition to a feeling of race, boundaries are also supported by religion in some groups; various Baptist and AME denominations for blacks; Catholicism for Mex-Ams; and various Jewish denominations for Jews.
As you can see in the graph, Hindus are very likely to marry each other, but it's more complicated than that. According to the GSS, only 19% of Indian Americans are Hindu (the modal category is Protestant--27%). Chinese Americans also lack a common religion: the largest group is people with no religion--44%.
But next is the interesting part: the unrecognized ethnic group--Mormons. There is a core whose family members have been intermarrying for several generations. Sure, they get lots of converts, but they lose a lot, too.
Many Mormons do genealogy and know their family trees and are always running into members who have a common polygamous great grandfather or some such connection. A large share of early converts were New Englanders, Englishmen, or Scandanavians, and their descendants have practiced endogamy for more than 150 years. Amy Adams who grew up Mormon has got the look.
American Mormons don't see themselves as an ethnic group, at least in the sense of blood. They have a strong sense of history, but it's a thought of as a religious connection. Theirs is a universalist religion. They won't end up like Jews. They think of those millions of Latino, Asian, and African members around the world as part of the family. They are very serious about marrying fellow members, but it doesn't matter much what a person's race is. Like with white Americans in general, though, marrying another white person is the rule (unless a missionary meets a brown hottie in the field).
The reality and future for the church seems to be a white core leading an increasingly multi-ethnic and multi-national organization. Like Catholics. Only on a smaller scale. And without fashion sense. And with bishops who get to reproduce like rabbits.