Mum's home takes payment monthly in advance. With regards what happens when the 'final day' comes, it depends on the circumstances. As a new company has taken over and changed the payment process then they will be providing a new contract to sign so ask for a copy.
You'll then need to check the specifics of the contract. I don't have mum's contract to hand at the moment but I think they ask for 28 days notice if someone wishes to leave the home, therefore there will be no overpayment of fees if you pay in advance.
If by 'final day' you mean when the resident passes away, you'll again need to check what the contract says as individual homes have different contracts but there should be some sort of refund paid, dependant on when the resident passes.