Let's define time travel through a moving worm hole. This is something I found that is credited to Kip Thorne (or the idea is)
Kip Thorne takes his wormhole and puts one end in his living room, and the other aboard a spaceship parked in his front yard. Thorne's wife, Carolee, hops aboard the spaceship to prepare for a trip. The two don't have to say goodbye, though, because no matter how far away Coralee travels, they can see each other through the wormhole. They can even hold hands, as if through an open doorway.
Carolee starts up the spaceship, heads into space and travels for six hours at the speed of light. She then turns around and comes back home traveling at the same speed — a round trip of 12 hours. Thorne watches through the wormhole and sees this trip occur. He sees Coralee return from her trip, land on the front lawn, get out of the spaceship and head into the house.
But when Thorne looks out the window in his own world, his front lawn is empty. Coralee has not returned. Because she traveled at the speed of light, time slowed down for her: What was 12 hours for her was 10 years for Thorne back on Earth.
Now, as Thorne and Coralee hold hands through the wormhole, they are each traveling in time. Coralee has landed on Earth 10 years after she left, and there she will meet Thorne, 10 years older. But she can still reach through the wormhole and find Thorne, who is only 12 hours older. Thorne can step through the wormhole and find himself 10 years in the future, or his future self can step back 10 years into the past.
At first this sounds possible, but this makes the assumption that the perception of time will be the same to the observer no matter what side of the worm hole they are on.
This breaks down if when Thorne looks in the wormhole he sees Carolee moving very slowly, not only her but everything on the other side of the worm hole. When she is looking through at Thorne's world, she will see everything moving very rapidly.
But she can still reach through the wormhole and find Thorne, who is only 12 hours older.
If his time was speed up when she looked in the wormhole, she could not find him only 12 hours older as he would have aged at an appropriate rate. If she waited 1.2 hours then went through the wormhole, he will have already aged apx 1 year.
Thorne can step through the wormhole and find himself 10 years in the future
If her time was slowed when he looks in the wormhole, stepping though after 12hours would be only minutes after she left. If he wants to step though when she arrives on earth he has to wait the 10 years.
This is where the fundamental assumption takes place
Thorne watches through the wormhole and sees this trip occur. He sees Coralee return from her trip, land on the front lawn, get out of the spaceship and head into the house.
If his view of her through the wormhole was slowed in proportion to her speed, this would never happen. In order to see her finish her trip and land he would have to wait 10 years, because everything on her side of the wormhole moves at an incredibly slow rate.
Now there is no reason to think time would appear to move at a constant rate from the point of view of both observers looking through the wormhole. Well unless I missed something. In fact we know the opposite ,which is that the perception of time can change.
As to what happens across the interface of the wormhole, that can get very sticky. There are many different ideas about that, I would suggest to be consistent in whatever you choose.
- Would there be visual lag across the gate?
This would depend on the speed of the wormholes relative to each other, as I describe above. It might also depend if it takes some amount of time to cross the wormhole. I could see how that is useful in a story. You would be at a slight disadvantage when crossing. Say it takes 15 minutes to cross, you would be working off information 15 minutes old.
- How would I treat the difference in gravity across the gate?
Gravity in GR is a feature of the geometry of space time. A wormhole is a topological feature of space time. I think one could argue that a wormhole can cancel out differences in those geometries (from one side to the other). If it couldn't then it probably wouldn't exist.
- Is potential energy maintained across the gate?
This is different then conservation of energy.
But If you talk about energy (just in general), I can see some things where stuff could get weird. Say you put a wormhole on the floor and it's other end on the ceiling (or some combination of wormholes). Then you drop a ball into it. It would pop out on the ceiling and fall picking up speed go back in the wormhole, pop out on the ceiling and fall picking up an infinite amount of speed. Basically you have a perpetual motion machine (you actual have better because it's accelerating). You could probably harness this unlimited gravitational potential to generate unlimited power. Honestly only way I can think how to square that is that you could put a limit on the energy something has when it passes through the wormhole. Maybe if it has to much energy it can deform space time enough to disrupt the wormhole.
This might have implications on moving a wormhole at high speed, because the more you deform space time the more unstable the wormhole would become.
In reality you can do whatever you want, whatever fits your need. You just have to be consistent and come up with a plausible reason. The big thing is thinking of other ways those rules can affect things, like I just did.