Yes... "...today's Democrats used to be Republicans, right? Back in Lincoln's time, he was a Republican, but the stances of that party are now represented by the Democrats? Is that what I'm reading? If Lincoln were alive today he'd be a Democrat."!!

It's not exactly a one-to-one flip by doing a "hey, I know what! Let's change names!" but the essential position of the parties as 'liberal' and 'conservative' has reversed. Lincoln ran against Stephen Douglas, who was a defender of the Dred Scott slavery act. The issues were aligned differently back then, but Lincoln better represented the values that 'Democrats' would espouse later. Some of the reversal was actually begun with Lincoln's election.
There were awkward periods when Southern Democrats..(Dixiecrats).. fought civil rights in the 50s & 60s, but they did it as 'southerners', and when Kennedy & Johnson pushed thru school desegragation, the shift was largely completed....except for the usual 'conservative' element who felt they needed the democratic label... but the conservative voting pattern... to get elected. And we still see a lot of that today.

It's not a simplistic division...but it is not fair to simply say "Lincoln was a Republican" and assert that proves 'Republicans' did a lot for civil rights.... "liberals" did a lot for civil rights, no matter what label/name was being used.