Sorry for sidestepping the question a bit, but I think there's a greater case to be made that the African-American Civil Rights Movement was sped along to far greater an extent by the Cold War, in turn opening up space for other social movements to emerge and advocate for their equal rights. Mary Dudziak argues this point in her work Cold War Civil Rights, (Princeton, 2000). Basically, her argument is that the geopolitical climate during the Cold War favored the expedition of the existing CRM that did in part emerge out of WWII, moreover favored by the Soviet propaganda machine and its response to legal segregation in the U.S. military. Given the struggle between East and West, and the incentives both sides had in gaining the favor of the rest of the world, the U.S. found itself in the difficult position of advocating freedom and human rights, while at the same time receiving criticism from abroad over a clearly hypocritical set of policies within its home soil. Dudziak further argues that the decolonization period, itself a consequence of the same geopolitical climate, placed more pressure on the United States government to deal with the issue of civil rights, particularly given the implications of the leadership of new African states visiting the U.S., and in some cases being subject to the same racism that African-Americans faced in the Jim Crow South (e.g. African diplomats being denied service in restaurants). She makes the argument that this, more than Kennedy's personal concern with the matter, led to the adoption of the Civil Rights agenda that remain part of his and LBJ's legacies. These aren't the limits of the evidence she employs, which ranges from numerous petitions to the U.S. attorney general for civil rights from foreign sources (including popular media and governments), the implications of Josephine Baker's planned visits to Cuba (and the fear this created), and the conservative backlash and attributions of civil rights to communist conspiracy against American society.

I highly recommend the work for the compelling case that Dudziak makes, while properly balancing international and domestic factors in the CRM, without doing much at all to deny the agency of the millions of people contributing to this movement. Moving back to your question, Dudziak does not (as far as I recall) make the leap from the African-American CRM to the other minorities that took part in advocating their own rights. Although, given the chronology of these events, I don't think it's inaccurate to argue that the African-American CRM did overshadow other struggles extant at the time (e.g. lingering first-wave feminism) due to its proving an obstacle to U.S. diplomatic interests during the Cold War, while at the same time opening up space for these other movements (second-wave feminism, Chicano movement, American Indian Movement, and Gay Liberation (and subsequent Gay and Lesbian Rights Movement).