I think that the varying branches of the Civil Rights Movement were directly tied to the expression of individual narrative and the need to articulate these in finding a greater sense of voice. Certainly, Dr. King articulated one aspect of the Black experience that spoke quite intensely to those in the South who were enduring that particular brand of discrimination. At the same time, there were those in the North who found the ideas of Malcolm X as particularly relevant to their own experiences from the Northern and Urban centers of the nation. College students found Stokely Carmichael as extremely meaningful to their own understanding. Women might have identified with thinkers like Moody, who argued that the process of liberation is as much gender as it is on racial grounds. The different branches of the movement has as much to do with the articulation of individual experience as anything else.
The most obvious "branches" of the movement to me are the Black Power part of the movement and Martin Luther King, Jr.'s part of the movement. King's included blacks and whites and emphasized nonviolence. The Black Power movement was more assertive and said blacks didn't need whites.
The movement had more of a King-type emphasis at first. But then, as true equality didn't come, there came to be more of the Black Power stuff.
Whites reacted much better to King's part of the movement, especially when it was asking for basic rights. They were much less sympathetic to the Black Power part of the movement and the parts of the movement that wanted things like more economic equality.