Few Democrats switched sides after the bill passed, but they did so knowing they were joining a caucus that supported civil rights.
Here is the dirty little secret Democrats use the race card to try to cover up. The Democrats split over Vietnam, not civil rights. After Kennedy and Johnson started the war, they lost the 1968 election to Nixon, who said he would end the war. Between 1968 and 1972, the party split, with the Kennedy/McGovern liberal wing seizing power. They even tried blaming Nixon for the war, which their media buddies pushed. The sun was setting on Scoop Jackson Democrats. Southern Democrats, who had always been strong on national defense and were mostly anti-isolationist, began migrating to the Republican party. Besides, many Southern Democrats were conservative and were lukewarm about the New Deal at best. But the Dems don't want to admit how they really lost the Roosevelt coalition by pushing the conservatives out of their party.
Just reading through the responses before I add my two cents, and I have to say yours is so far the best, but you don’t go far enough back in history.