Can gravity in principle be made to flow through a "wire"? Electricity can be made to flow preferentially through a conductor such that a potential difference can be exploited and it can be used to do work remotely. In principle could an analogous "gravity conductor" be made that would allow a difference in gravitational potential to be preferentially directed along a particular route. If not, what is the fundamental difference between gravity and electromagnetism that prevents this from being possible? 
 A: Electricity does not flow. Charge carriers (electrons) do if subject to an electromagnetic field. When all the charge carriers have flown from $A$ to $B$ such that they produce an electromagnetic field that evens out the initial one, then the system is in equilibrium, which is equivalent to saying that the two points of the conductor have reached the same potential.
Equivalently gravity does not flow but mass carriers will. This means that whenever you place a mass carrier into a gravitational field it will move according to the direction of the field. You can imagine the same experiment and eventually reach a state where the masses have flown from one end of the gravitational conductor to the other in order to even out the initial gravitational field, namely the two boundaries of the conductor share the same gravitational potential.
This said, there is indeed a big difference between electromagnetic and gravitational fields, which is the fact that in the universe there are two types of electromagnetic charges (which we refer to as $\pm$) whereas there is only one type of gravitational mass: no negative masses exist. This is the key factor that generates all the differences, even in quantum field theory, between the gravitational field and all the other interactions present in the universe.
