In the prose that crosses my desk I regularly find references to the Civil Rights Movement, always capitalized.
The Associated Press Stylebook does not address the issue, which suggests that AP doesn’t think capitalization of the term is a matter it needs to address; the Chicago Manual of Style specifically indicates that it should be lowercase.
Merriam-Webster.com does not address the point, and Webster’s New World, fifth edition, lowercases civil rights movement, though noting (Aha!) that it is often capitalized.
I checked the Corpus of Contemporary American English, which shows a mixture: civil rights movement, Civil Rights movement, Civil Rights Movement—though the lowercase term is overwhelmingly dominant.
The context in which I see the capitalized term is invariably the African American struggle for equal rights in the 1950s and 1960s, or events or circumstances rising from that era—a shorthand for King-era events.
I am not happy with that narrowness of focus, because the efforts of women and gay people to gain equal rights, up to and including the present day, must surely be components of a civil rights movement. And I am reminded that the American Civil Liberties Union, founded one hundred years ago, has been in the movement for the long haul.
So I think my choice to lowercase it whenever I encounter it is amply justified.