One theory is that the end ("end") of segregation began and hastened the decline of both the church as it did Black-owned businesses, but I think it is more than that. Because, as Aaron once said, it's not like white people came flocking to Black business after the signs came down...But in the church realm, it was a little different, at least in some areas of the country, particularly the West Coast. Sometimes, Protestantism is right in line with the concept of branding. In the postwar period, there were some churches that had had enough of hierarchical polity. They started to form around the concept of "non-denominationalism".
Out here, a lot of these "non-denominational" congregations were somewhat integrated, considering, as they saw themselves as introducing a new theological paragidm, based on a sort of nostalgia for the early church, but with the most modern of trappings. Whereas traditional Catholics and "fundamentalists" are more likely to shun media and new technologies as they emerge, viewing them as a hindrance to one' spirituality, the Evangelicals/Charismatics/neo-Pentecostals jump on at least a narrowly optimistic vision of technology to suit their ends of spreading "The Truth". Let's put it this way: they are not afraid of the concept of the "network and the circuit". (This is something I am trying to develop in my own research, part of a larger project on Gospel music). As far as so-called "multiculturalism" was concerned, these Holy Rollers and wonder-workers were light years ahead of anything on the Left, who finally began to catch up on a rhetorical level, some time in the late 80s. And all of Pentecostalism has its roots in the Black church, anyway, beginning at the turn of last century (1906 to be exact), at a church meeting called the "Azusa Street Revival" (here's a piece on "Hispanic Pentecostals").The Pentecostal groups, neo- and classical as I call them, are theologically quite liberal, in that they believe that "the Spirit" can work through anyone, regardless of race or gender, and they claim a rather universalist idea of language, with their emphasis on "speaking in tongues". Although they have no theological or theoretical problem with female preachers, they are otherwise socially conservative.
To be fair, for the climate and time they got their start, they were farily progressive in some ways, while being reactionary in others. Tent revival meetings on the evangelistic circuit, depending on locale, were often segregated, if that was the custom of the county or city. Some neo- and 2nd/3rd generation Pentecostals, wise to the future, put a stop to that; in fact famed televangelist Oral Roberts once wrote in one of his bios that in 1948, it was his tent meetings in and around Oklahoma that were among the first to desegregate their traveling congregations. About that same time, the Civil Rights Movement as it is commonly known, was in full swing, and as we know, a lot of history waited in the wings. The Armed Forces were declared desegregated in that year, as were federal jobs for civilians. It would take the white Methodists to end its race-segregation policies, which it had set in motion in 1939. Jackie Robinson had become a Dodger the year before.