Since the south was a one party situation, that meant all interests were voiced through the Democratic party. Those interests were of course labor, business and agriculture primarily. Suppressed turnout meant that only the elites really had any political power or influence as the turnout was low in many states like Mississippi. It varies by state since some states had populist traditions that flowed into a more pro-New Deal stance, where in others that was not the case. A good example to look at was Texas, where as late as the early 1960's, you had the Progressive and Conservative business factions warring with each other and JFK's trip to Dallas was part of a multi-city swing to both mend fences and to boost his standing in the state.
As time went on, the business wing would start to drift towards voting like their counterparts in the rest of the country, starting with Eisenhower and generational politics played a big role as you got more removed from people who personally witnessed the Civil War and its aftermath. Anyone born prior to the 1930's would have probably grown up hearing someone raving about Yankee Republicans burning and decimating the South. Afterwards that begins to fade, tribalist opposition to the GOP goes with it and soon practical policy concerns on issues of business, civil rights and other issues take far greater priority than legacy voting.
Texas is a fascinating example. Between Coke Stevenson, Allan Shivers, John Connally, Lyndon Johnson, and Ralph Yarborough, there was a wise variety. Shivers, I believe, was conservative on every issue but civil rights.
