When exactly did this "switch" occur? When it benefited democrats i'm sure! Byrd was a democrat representative and senator from 1953 until 2010, so the fact that he was a racist democrat when he started really meant he was a republican? And when he apologized so he could further his career he then became a "real" democrat, but the whole time he was called a democrat? The Republicans that supported the CRA were actually democrats and democrats that fought it tooth and nail were actually Republicans, so the switch happened pretty recently (historically speaking), right after the CRA was passed?
No, the there was a gradual transition, most notably, during the T. Roosevelt, Hoover, FDR, and JFK administrations. Presidential candidate William Jennings Bryan (of Scopes Monkey Trial fame) is cited as one of the first that started the parties to revamp their platforms.