This is probably along the same lines of the perpetual question about how long you remain a Toad if you miss a turn during your time as a Toad. But I'm going to ask my question nonetheless.
Character 1 casts Destroy Magic at the end of his turn. The spell remains in effect until the start of his next turn.
Character 2 rolls a 1 for his movement and subsequently moves the Reaper onto Character 1's space. Character 1 rolls a 4 with the Reaper, meaning that he misses his next turn.
Does Destroy Magic remain in effect until Character 1 actually takes a turn, even if it ultimately means that the spell is in effect for two rounds?