The GOP was the Civil Rights party for 100 years. They were the business party, but they were the big government business party. V. O. Key once wrote how, the long period of Republican rule was due to more than the Moral Cause of the Union and the Victory that entailed. "Sentiment clothed bonds of substance," wrote Key. And those bonds of substance included protective tariffs for Northern industries (but not for Southern crops, save sugar), military pensions (for Union soldiers, not for Confederate soldiers), subsidies for railroads (which forged their routes to direct the flow of commerce away from the South, and free land under the Homestead Act. Whatever you want to call these measures, they are not the acts of a "small government" party.
It's difficult for us to understand today, but the ideological debate over the role of government was broadly between laissez-faire on the left and government intervention *to support big business* on the right. Before New Liberalism in Britain, Bismarckian welfare in Germany and the Progressive Era in the United States, the idea that the government could be used as a tool to remove inequalities instead of perpetuating them was one for crankish philosophers. The welfare state did not exist and the income tax did not exist, so the central government was more or less a war-making machine that funded itself through regressive taxes on the working class like the tariff. This is why Jefferson, Jackson, Grover Cleveland were on the left in their day, opposing government because it was seen as only ever operating by exploiting the poor in the interest of various power elites and monied interests.
In this context, one can understand how a progressive policy like Reconstruction and federal enforcement of civil rights became associated with the right wing party and why the Liberal Republicans bolted. Democrats saw federal troops occupying the South as one with a general policy of corrupt, hierarchical central government intervention crushing the little man.