You'll probably get an answer from a historian soon enough (I'm just a reader like you), but in the meantime there's a few links in the wiki that appear to be related to your question:

Changing role of Republicans and Democrats

• Myth or Fact: Did the U.S. political parties switch platforms? by u/erissays
• Can someone address a brief history of Democrats vs Republicans, specifically the change in Dems from the early 1900s being against civil rights to a more progressive party in the 50/60s leading much social change in the U.S.
• It seems that during the Civil War and reconstruction era, Republicans were liberal and Democrats were conservative. When and why did this change?
• How and when did the Republican Party transition into the Democratic Party we see today (and vice versa)?
• I've heard that the American South used to vote solidly Democratic. What happened, when did this change?
• What was the relationship between government and business during the mid-20th century? by /u/tayaravaknin
• Why was there such a huge shift in the core viewpoints of the Republican party in the past 100 years? by /u/samuel_gompers