They have to exhaust the appeals process first.
That means requesting appeals from numerous higher courts and asking elected officials for stays, pardons, etc.
Since a death sentence is final, they usually look at these requests very carefully, and examine all the evidence, original trial, everything to make sure that there are no indications that the person isn’t actually innocent. Because if they are, there are no take backs on an execution.
An appeal can be granted for all kinds of reasons. Maybe the cops mishandled evidence, or a higher court judge disagrees with a ruling the trial judge made, or new technology allows existing evidence to be interpreted in a new way. Anything like that can be grounds for an appeal.
Any appeals that are granted have to go to trial. If the person is still convicted, they stay in prison and petition higher courts for more appeals. If they win, they go free. Eventually, they will exhaust all of their potential appeals and will be executed, but that usually is a decade or more after they were originally convicted.
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