The mechanism is for the voter to notify the old location they have changed residence and are no longer eligible to vote there. The founders envisioned an educated, informed, honorable citizenry that cherished their freedoms to the point where personal honor and integrity would undergird and protect our most precious right- the right to elect our representatives!
Without self discipline and honor our system cannot stand, people don’t obey the laws because they have to. They choose to obey and live honorably. If we are not a moral people under God, we cannot last as a free people.
"Our Constitution was made only for a moral and religious people. It is wholly inadequate to the government of any other." --October 11, 1798 John Adams