It seems like a problem of states trying to pass laws that control things outside their borders. The jurisdiction of Louisiana courts is Louisiana.

I mean it would be absurd if an anti-death-sentence state started trying to extradite the executioners working in pro-death-sentence states for murder, right?

If the executioner did their work in the anti-death-sentence state it wouldn't seem to be absurd, no. E. g. if they had pulled the cord that activated the electric chair remotely from a pro-death-sentence state (tele-execution ... sounds very BlackMirror).

I’d expect that to result in a very confusing court case. Fortunately, despite all the other messes going on, we haven’t tried anything that silly.