The Dixicrats left the democrat party in 1948 and they left because they favored segregation, Jim Crow, and white supremacy and hoped that they would find sanctuary in the republican party. Affirmative action, school busing, bra burning hadn't happened yet. They formed a new party called the States' Rights Democratic Party. One of their central planks said,
We stand for the segregation of the races and the racial integrity of each race; the constitutional right to choose one's associates; to accept private employment without governmental interference, and to earn one's living in any lawful way. We oppose the elimination of segregation, the repeal of miscegenation statutes, the control of private employment by Federal bureaucrats called for by the misnamed civil rights program. We favor home-rule, local self-government and a minimum interference with individual rights.The States' Rights Democratic Party didn't last too long and some returned to the democrat party with others drifting to the Republican party.
It is true that the Republicans favored less government than the democrats, but only by a matter of degree. It was conservatives operating within the Republican party who truly favored smaller, more responsive government, but the Dixicrats only wanted the federal government to leave them to continue their practice of oppression.
When I refer to Dixiecrats I am referring to Southern Democrats, not the party that was started in 1948. Southern Democrats were still referred to by no-nothing northern politicians as Dixiecrats.