How to best make a prepayment on your mortgage

One very important point I never hear discussed is what the lender does with the extra loan payments borrowers make. If you pay extra on your mortgage before it’s due, does the lender require that the next payment be paid on schedule or can you skip monthly payments corresponding to how much extra you’ve paid? It would seem that many payoff amounts calculated at some point in the future by the lender might go back to the original recorded security deed and original amortization schedule with the extra payments overlooked.