This is a very interesting question. On several levels, the desire for independence in colonized nations and the desire for articulation of voice through the Civil Rights Movement shared some powerful convergence. The idea of freedom is something that was appropriated by both. As leaders of the Civil Rights Movement in America sought to grasp "the fierce urgency of now," they looked to similar movements in Africa and Asia to help bolster the case that the "now" was the time needed to embrace change. Additionally, both movements felt some level of kinship towards one another as the struggles of people of color had a universal feel to it. Finally, the movements in Africa and Asia that overthrew colonial rule did so reaching to the past of America, to a certain extent, as it did the same thing with its relationship to England.
These kind of fed into each other -- sort of a two-way street.
Decolonization in Asia generally started to happen before the Civil Rights Movement. As these Asian countries became independent, they became pawns in the Cold War. As the US and the USSR competed for those countries' loyalty, civil rights became an embarassment to the US. US treatment of non-whites made us look bad in the eyes of these new nations.
Decolonization in Africa generally happened around the same time as the Civil Rights Movement. Africans took heart from the US movement and it helped encourage them to push for independence. At the same time, African Americans became more aware of and into their African heritage as these nations became independent.