I can answer in broad terms. By the time of FDR, if not long before, the Democrats were really two parties joined in a national alliance - the progressive New Dealers in the North, Mid-West and West and the anti-Yankees in the South. (The Years of Lyndon Johnson, Robert Caro). The Southern Democratic party was extremely reactionary and racist but the South would never vote for the Republicans because of the Civil War. Caro describes how the Southern Democrats and Northern Republicans had a lock on The Senate for decades preventing reform. FDR even went to the lengths of intervening in Democratic party primaries in the South in the 1930s to try and obtain more progressive candidates - without success, and obviously, to later problems in Congress (The End of Reform, Alan Brinkley).

This situation only started to change after the passing of the Civil Rights Act by Johnson (and, by the by this probably would not have been successfully passed by Kennedy - see Caro for the tactical genius Johnson demonstrated) - the South began to think the Democrats did not have their interests at heart. This leads to Nixon's "Southern Strategy" - i.e. the Republicans trying to pick up the Southern states in the Presidential election and that leads to a long period of Republican presidential dominance that only starts to be reversed by Clinton - a Southerner, of course; and one with a forthright position of on racism as part of his politics of the 'New South'.

TL;DR = the South never forgave the Republicans for the Civil War until LBJ passed the Civil Rights Act.