FDR started their migration, LBJ sealed it, but I don’t think it’ll last for the 100 years he projected.
You have the right take, the switch was made in 1936, not the 1960s.
The switch was a complete and instant reversal, in 1932 the black vote was like it always had been, FDR got his 23% while the GOP got its over 70% which were typical numbers, but suddenly and permanently the black vote reversed itself in 1936 and FDR got over 70% and that was it.