Two things come to mind when the term "break-in" is mentioned for any device with moving parts. Both have acquired mythic status.
The first is avoiding damage or undue wear when a new device is first used. New car engines were supposed to be run gently during a break-in period. No auto manufacturer seems to claim this anymore, but the myth still appears. As far as I know, no speaker maker has ever claimed that new speakers should be run more gently than normal to avoid damage. Occasionally new buyers wonder about this, but their caution is no different than the understandable fear of breaking a new toy.
The second myth has it that one should avoid judging the sound of a speaker until it has been "broken-in" for a number of hours. Depending on who is talking, the time span for this varies from several seconds to hundreds of hours. I have never witnessed even a poor, much less a convincing demonstration of a speaker's sound changing as a result of break-in. It is said that manufacturers and sellers take unfair advantage of this myth to avoid refunds of items recently sold.
If speakers don't need to be babied when new (as car engines may have been many years ago), and if speaker "break-in" can only be measured, and barely or not at all heard, why are we talking about it?
Use new speakers as they were intended. And yes, start judging their sound right away.