Speaker designers will usually run new woofer samples (and cone midranges too) at fairly high voltage for 20 to 40 minutes at or below their measured free air resonance using an audio oscillator. This will change the Thiel-Small parameters of the drivers very slightly. For example, the free air resonance of a 6.5" woofer might change from 48Hz to 45Hz. A 4" midrange might change from 150Hz to 145Hz. Not much.
Tweeters never ever need break-in. If you don't like the way the aluminum dome sounds, it isn't going to change with use. What might change very, very slightly is the overall system balance because the woofer's free air resonance drifts ever so slightly lower.
Other than that the whole break-it-in-and-it-will-sound-better line is nonsense routed in the psychoacoutics of the situation. It simply means that your brain adapts and becomes more tolerant of the bad sound over time.