Hi, law student here. It is sort of a free speech issue. Typically voting rights challenges are brought under the 1st and 14th amendment. Basically, you say, “The law did not give equal protection (14th amendment) to my right to free speech/freedom of association/ petition the government for grievances (1st amendment).
The law is immeasurably messy here because, in all honesty, it was sort of made up because of judicial overreach. Basically, if our current justices/line of legal reasoning were used in the 1960s, there would be rampant voter suppression. Chief Justice Earl Warren kinda invented voting rights because he thought the current system was fucking bullshit. He was right, and Warrens decisions are basically the foundation of modern America. I think he oughtta basically be considered a founding father of the constitution because he sort of reinterpreted the entire constitution in order to make society good.
The problem is that he didn’t really… do the job of a judge. What he did was definitely good and right. But it wasn’t really his job to just declare that voting rights matter more than anything else. So nixon and reagan’s judges came in and basically said, “Everyone oughtta give a hand to Warren for helping America so much! But now we’re gonna stop doing that and dial everything back a notch or ten.”