to go on ahead, or to lead someone out to the front: motion to the front, whether going before others yourself or bringing another forward
The word joins two ideas: "before" or "forward," and "to lead" or "bring." Its basic picture is motion to the front. Sometimes the subject moves out ahead alone, going on before others; sometimes the subject takes another person and brings them forward, out to where everyone can see.
That gives the word two directions. Going ahead yourself is the more common one: a guiding star moves out in front of the travelers, crowds stream on ahead shouting, and the risen Jesus tells his followers he will go before them into Galilee, the shepherd leading the flock to a meeting place. Bringing someone else forward is the other: in the book of Acts it repeatedly means to lead a prisoner out, hauling Peter from his cell or bringing Paul before a ruler to be questioned.
From moving through space the word stretches to moving through time. A "foregoing" commandment is an earlier one, now set aside; prophecies made earlier still point ahead to Timothy; and sins can run on ahead of a person all the way to the place of judgment. One last use turns the picture into a warning: in 2 John, someone who "runs on ahead" and does not stay put in the teaching of Christ has gone too far, straying past the faith rather than growing in it.
A plainer relative, προπορεύομαι, also means to go on ahead, but only ever of the one who is moving; when the stress falls instead on leading someone out of a place, Greek reaches for ἐξάγω.