The Democrats co-opted the credit for the Civil Rights Act of 1964. But if you go back and look at the history a larger percentage of Republicans voted for that than did Democrats. But a Democrat president signed it so they co-opted credit for having passed the Civil Rights Act of 1964 and the Voting Rights Act of 1965.
Martin Luther King Jr. didn't carry just a piece of cloth to symbolize his belief in racial equality he carried the American flag.