Michigan law states that the voting adddress and driver’s license (or state issued ID card) address must be the same. If you get an out-of-state drivers license, you just forfeited the right to vote in Michigan. Also, if you register using an address that doesn’t work two weeks later when the voter card is mailed, it seems to me you never lived there.
Should people who happen move in October forfeit the right to vote in that year's presidential election? That would hardly seem fair.
IMHO, the proper approach would be for the person who moved in October to cast an absentee ballot; on Election Day, the state where he had previously lived should check with his new state to confirm that he was not yet eligible to vote in its elections (the simplest way to handle that would be for the new state to include on its registration record the date when he'd become eligible to vote there).