To expand on what Destiny Warrior said:
As everyone else explained wormholes are literally a fold in spacetime. The diagram from A Wrinkle in Time illustrates it adequately; what you're doing is bringing two different sections of space together, from what I understand.
Now I'm going to have some fun and get into a bit of particle physics.
Cookie, you mentioned that we can replace gravity with whatever other field force we want, so in order to create a slightly simpler explanation I'm going to replace it with the electromagnetic force since it's simpler to describe in terms of particles.
So if we shot light through a wormhole it appears at the other side nearly instantly right? What light is is a wave/particle thing which is a photon when viewed as a particle. So this means it's quite simple to shoot photons through a wormhole and get one out at the other side. Well, according to the field of quantum electrodynamics, the electromagnetic force is just the interaction of the exchange of virtual photons between charged objects. In other words, it's that objects with electric charge send out photons between each other to create this field force that we call the electromagnetic force. So if instead of gravity, we place a strong electromagnetic force next to the wormhole, then the field will logically extend out and hit the wormhole. The virtual photons (again basically what the force is) that comprise the field force go through the wormhole, meaning that the force is carried with them. TL;DNR: The way that field forces work on a small level they should be able to leak through a wormhole and come out the other side relatively undisturbed. Thus whatever part electromagnetic field that hits the wormhole gets sucked through and pops out at the other side with whatever strength or intensity it had. So if we were to throw a charged object through this wormhole that is next to our electromagnetic field, on the other side the object will still feel the exact same force as it would if it were sitting right where the entrance to the wormhole is.
As gravity is a field force, even though its exact mechanisms are vastly different from our conventional idea of a field force, I imagine that its field of influence should leak out through the wormhole the same way as the electromagnetic force, meaning that if we throw our object through the wormhole then on the other side it shouldn't have gained or lost any potential energy. Thus some kind of perpetual motion machine is out of the question. Since the object is still in some kind of gravitational field at the other side of the wormhole, even if it's far from earth, it won't have gained any potential energy (D:). So if you guys's explanation for the loss of energy the object should gain from the sheer distance is right, this would maybe explain how it's conserved.
If we were to then grab our object on the other side and pull it further out away from the leaking gravitational pull, it would require energy to move it away the same way it requires energy to escape gravity's pull here.
Edit:
You can kinda see what I'm talking about if you imagine some kind of gravitational or electromagnetic or other field right next to the bottom end of the wormhole in this picture. The way the field works, it would naturally "leak through" and the field would come out the other side in the same direction it was going relatively undisturbed, as though nothing had happened at all. The explanation MrIndigo had that there is a loss of energy is shown in that diagram as well (all the "negative energy" I suppose).