I have a magic continuity question…

At the end of HBP, Dumbledore’s freeze charm on Harry broke when Dumbledore died. So given that logic, how does the magic at Hogwarts castle work? Do the teachers have to reperform the protection spells every time the one who cast them dies? The sorting hat was made by magic, (I’m assuming Godric Gryffindor himself performed it) so how does ensuring legacy magic still work after the spellcaster is gone?