Might some connection have been loosened, but not entirely disconnected, at/in the problematic speaker during the move? If so, it could be that some cabinet-wide vibration occurring at moderate volume levels is causing the poor connection to make and break very rapidly.

Another guess: something in a crossover is allowing a small bit of lower-than-intended frequencies to travel to the tweeter. At lower levels, where this does not cause distortion, you wouldn't normally notice because these same sounds are also coming from the drivers intended to reproduce them. Once the levels are high enough, the tweeter is mildly over-driven by the unintended frequencies, but not yet damaged.