In the case of a speaker that provides separate low frequency and mid/high frequency terminals (that are truly separate; there may be some designs in which they are not), the crossover will minimize or eliminate the low frequency content of the current that flows in the cable going to the mid/high frequency section of the speaker. It will also minimize or eliminate the mid and high frequency content of the current that flows in the cable going to the low frequency section of the speaker.
The voltage appearing at both pairs of speaker terminals will be essentially the same, however, and will be comprised of all of the frequencies that the amplifier is putting out. Although that is arguably irrelevant, as current flow in each of the cables is arguably what matters.
Regards,
-- Al