Mileage varies depending on how much you like teen drama and/or Karate Kid but it is the gold standard of how old shows and movies should be adapted.

The lore is intact but events have moved into the future in a logical way. Characters keep their attributes but are more mature/jaded. Introduce new characters through the old and gradually shift focus if they're interesting enough. Bring back side characters and develope their personality. Treat the history with respect as time moves it forward.