Lincoln was a Republican and had support in the north. After the war the Dem party was greatly weakened. The Republican party was in part built around the power of industry while the Democrats were built around the plantations and the interests of farmers. There was a lot more to this obviously.
In the north the Republican party was very dominant and started to alienate Catholics and some other groups by pushing things like prohibition. The Dems were also able to gain ground by supporting unions. Economic turmoil tends to push people towards the other party and the Republican stranglehold on power only lasted as long as the good economic times did. In the end the way the parties defined themselves changed drastically from the end of the civil war to Nixon's southern strategy.
It is pretty typical for a political party that is losing to start to define themselves by appealing to those looking for an alternative. This is what the Dems eventually did after the war and is what the Republicans did in the 70's.
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