This is an incredibly complex question, and any attempt to simplify the answer by either insisting that Southern Democrats were just racist versions of Northern Democrats or by acting like they were all right-wingers who just called themselves Democrats for ancestral reasons should be looked at VERY suspiciously. It'd take too much time to go way into this now (mechaman and NC Yankee should be required reading for anyone interested in this topic), but you have to remember that the South was a de facto one party region. Whenever you have a one party region, you will (almost by definition) most likely encompass a wide range of ideologies in one party. Sure, you had very right-wing Democrats like the Byrd family in Virginia and Strom Thurmond who were only Democrats because it was a necessity to hold higher office, but you also had very clear progressives like Long and Black whose only supposedly "conservative" views were that they were racists and supported segregation - something that I don't even think is inherently "conservative" or fits on a left-right scale at all. The main unifying factors in the Southern Democratic Parites were a defense of segregation, an ancestral tie to the party that was most aligned with the South in the Civil War and a desire to have the region unified when it sent its Senators to D.C., as they believed this was the best way to look out for the South's interests. As the South was beginning to become a two party region in the 1950s, you had people like Ross Barnett claiming that the worst mistake the South could do was allow any Republicans to take advantage locally of the tension between Southern Democrats and Northern Democrats, as it would only eventually hurt the South. In support of a Democratic gubernational candidate after he had left office (from his Wikipedia page):
Barnett employed his fiery rhetoric when he urged his state's Democratic voters to "push out this Republican threat" and added that he was "fed up with these fence-riding, pussy-footing, snow-digging Yankee Republicans", a reference to northern transplants coming into Mississippi.
Many of these transplants were bonafide conservatives (not just Republicans) moving South, so it wasn't simply a question of ideology. I think I did a fairly decent job of explaining why it's very simplistic and dumb to just think "segregationist Democrat = conservative" in this response:
https://uselectionatlas.org/FORUM/index.php?topic=285458.msg6141852#msg6141852In short, Southern Democrats are far less ideologically homogeneous than many like to paint them as. They weren't all "racists with left-of-center economic views" (though I would argue that might actually be a plurality at many times throughout the Twentieth Century), and they certainly weren't all "conservative DINOs." If you simply can't separate "segregationist" and "conservative," I guess that is your issue, and this discussion becomes a lot less interesting. However, I think it's a lot more complicated than that, and it truly is a fascinating party dynamic.
*Paging NC Yankee*
