Oh guess which party did that? The Democrats unless you buy the "southern strategy" nonesense.
Whether or not you believe in the existence of the Southern Strategy as a deliberate act, it's pretty clear that the South has steadily shifted toward the GOP since the 1970s, particularly escalating in the 90s. Coming after a very public schism where many Southerners tried to create a third party rather than vote Democrat or even Republican, just for their regional third party candidacies to fail to do more than win some of the Deep South states.
Arguably, what's happened is that there used to be an alliance between the urban working class and unionizers and agrarian Southern whites to oppose the Republicans' traditional focus on supporting Big Business. But the urbanites, especially the West Coast and Northeast ones, embraced a progressivist plank that included racial equality (although not always easily or whole-heartedly), and many Southern whites saw this as an attack on Southern social structure, so they started breaking with the Democrats over it. As was seen with the aforementioned third party runs in 1948 and later in the 60s.
Modern Democrats have few links left to the old coalition, so they feel free to wave the bloody shirt of the Civil Rights movement (aided by JFK and LBJ both backing civil rights legislation, so they have that going for them) since they don't consider the Southern Democrats to be "their" predecessors.
The GOP, with its continued favor toward
laissez-faire-esque policy and limiting government, and especially federal, power, are attractive to the traditional Southern viewpoint of state's rights, so it's no surprise that having lost the Democratic Party to the urban progressives and liberals, they've turned to alliance with the GOP.
Which, of course, means modern Dems feel free to make such attacks on the GOP on the matter of voting rights and accusations of plans to disenfranchise minorities and the poor.