Let's say that we made a wormhole time machine where one mouth was held in my backyard right now in the present time, and the other mouth travelled into the future for 100 years. What happens to the mouth of the wormhole that is in my backyard right now over time? Will it still be there in 10 years? Obviously, the "future" mouth existed for 100 years and the one in the "present" had to be able to accelerate to keep it from advancing in time, so if the mouth in the "present" can start travelling into the future after the creation of the time machine, then how do you know what time you would come out if you went into the relative "future" one? Doesn't the "present" mouth have a world line, or move in space and time itself?
Wormholes defined using General Relativity don't work in that way.
As far as most realistic scenarios go, they may be able to exist with both ends positioned on the same space-time, meaning that the current time near each of the wormhole end points is the same.
All of the time warping effects would be simply the result of time dilation caused by the space-time shape necessary around a wormhole. That shape would be very similar to a black hole. This means that there would be a huge time slowdown effect near such an object. So in a sense, if that effect somehow wouldn't stop the passage of time entirely as you approach the wormhole endpoint (as it most likely should when approaching a black-hole), and passage was possible, then besides traveling to a different point in space-time through the spacetime tunnel being the wormhole itself, you would effectively time travel into the future. The problem here is that if you were to go back, you could only go back to the same point in space-time as before, but in an even more far future. It's often stated that traveling faster than the speed of light could make you effectively travel back in time, but that still purely assumes that you can't travel faster than the speed of light in the present (meaning that the additional speed would be gained by gaining not only speed through space, but also gaining "negative speed" through time (and that brings up all kinds of problems)). In a more "reasonable"(haha) approach, where we would not obey such a limit, if you were traveling faster than light, than time would most likely stop for you for the time of the travel (causing you to effectively travel through time into the future as in previous examples), but you would be able to overtake any light-speed emissions, so in a sense you could peek into the past, by enabling yourself to, for example, quickly travel in 10 years, 60 ly away from Earth, and then turn around, move a bit sideways, and observe light and all light-speed emissions from Earth that are 50 years older than the moment at which you left Earth (and if you came back in the same way in total 20 years would pass, so there would be still no actual backwards time-travel through spacetime). This is as close you could get to traveling back in time, while it already requires breaking the fundamental, known limits of spacetime, so I think it's safe to assume that actual time travel into the past in a single point in space will be never possible.
So in summary:
- Considering that the ends of a wormhole should look similar to a black hole, there's no reason really why a wormhole's ends couldn't move around (there's also a problem of how exactly can you define any object in spacetime as truly static since it depends on the observer?).
- As far as GR goes, if a wormhole was possible it would practically be a time machine that only goes into the future, similar to simply traveling near the speed of light.
Ps. To clear up the "one end traveling near the speed of light" does not mean that the present in that end would change or stop. The time dilation effect affects matter, not the spacetime itself. So this way would not introduce any time difference near the ends of a wormhole.