It’s a very complex question. You might find the following article on statistical studies of racial diversity among US churches (of all denominations) interesting
encyclopedia.com/doc/1G1-71949662.html
Catholics actually appear to do a good bit better job than mainline Protestants in this.
*Ironically, the poorest record on diversity–only 2 to 3 percent mixed on average–belongs to historic Protestant churches, which were among the first to trumpet the ideal of integrated congregations… Catholic churches “are almost three times more likely to be multiracial than are Protestant congregations” because the large parish boundaries normally embrace several neighborhoods, he said. Yet the Congregations Project found less socialization and interaction between ethnic and racial groups in Catholic parishes, which often have separate masses for different language groups.
The more integrated churches among Protestants usually were the more theologically conservative, non-denominational congregations. Overall, the study found that only 7 percent of Protestant congregations nationally could be called “mixed.” …
Looking at independent, nondenominational churches in the Hartford study, research associate Scott Thumma said that nearly 25 percent of them had a substantial mix in which the dominant racial group was no more than 60 percent of the congregation. “A plausible explanation is that denominational labels create a cultural expectation of whether they are
white' or black’ churches,” said Thumma, noting that, like Pentecostal churches, nondenominational congregations have more contemporary worship styles and flexibility.
Floyd “Butch” Gamarra, missioner for multicultural ministries in the Episcopal Diocese of Los Angeles, put it another way: “In a lot of mainline churches the issues are race and class.” Liberal churches present the theological idea that “the world is supposed to be a rainbow,” but the upper-middle-class church members “want to be cerebral” about it, Gamarra said. The Pentecostal and independent churches “tend to attract more working-class people who are in the same social, economic class,” Gamarra said. “The mix is a lot easier.” *