Putnam and Campbell found that 90% of American congregations are ethnically homogeneous. There is still a strong ethnic foundation to religion, most especially for Jews, Black Protestants, and, to a lesser extent than a generation ago, Catholics.
The least ethnically homogeneous congregations were found among Mormons and evangelical Protestants. This is somewhat ironic since Mormons famously resisted admitting black men to their priesthood until the 1970s, and conservative Protestant sects were the core of the religious resistance to integration and black civil rights a generation ago.
However, the whole nation has enjoyed a sea change in racial attitudes. Religious people are now against racism and for ethnic diversity, pretty much across the board. The remaining racists tend to do their organizing outside of religious networks.
Moreover, it makes sense that Mormons and evangelicals would be creating congregations that are increasingly diverse by ethnicity: these are the faiths that most evangelize new people into the faith. And, for the same reason, Mormons and evangelicals are the least ethnically based of major faiths, because what holds them together is common faith, more than a common background. Evangelical megachurches, in particular, have made a concerted effort to evangelize beyond their white base, which has paid off in the past decade or so.
The growing points of American religion are getting less and less segregated, and the younger generations are more and more likely to value ethnic diversity. This bodes well for the future.