"To push a trip forward/back"

How about: We had to leave earlier than (we had ) planned. or
We had to reschedule to an earlier date.

No because like many things in language the idea comes from a perception of a physical reaction. The event is coming toward the person, not the other way round. You know this because people can stand still but time does not. So, when you push something back it is like you are pushing back an object coming towards you.

Maybe.

another alternative to indicate an earlier date is
TO MOVE UP THE DATE

Why no, Simon?

Sorry I think I was disputing something Tom said.

No problem.:-)