In theoretical physics, the issue of route-dependence deals with whether a selected differential between two points is taken as absolute, or as being partly a function of the route along which comparative measurements are taken. It usually applies in discussions of gravitational potential or related effects such as gravitational redshift.
If light-signals are exchanged around one side of the hole, in the equatorial plane, where the adjacent section of event horizon is moving roughly in the direction A→B, then frame-dragging effects should make it easier for light to move with the horizon's motion than against it, and the measurements should show B to be downhill of A.
If we repeat the exercise with light-signals sent around the other side of the hole, the resulting anisotropy in the speed of light will now act in the opposite direction, and B will appear to be uphill of A.
At first sight this seems like a "bad" result as it allows energy to apparently be obtained "for free", because we can surround the black hole with a circular track, and allow an object to repeatedly fall "downhill" around the track from A to B and back again, extracting energy each time, and thus violating the principle of conservation of energy.
On further examination, since the energy that we extract from these objects should create a mutual dragging effect on the spinning black hole and fractionally slow its rotation, the energy removed corresponds to the reduction in the rotating black hole's rotational kinetic energy. Energy conservation is not violated.
Adjacent local gravitational potentials can be measured along specified routes, but one could surround a rotating body with three satellites, A, B and C, and say that according to the signals sent between adjacent objects, A>B, B>C and C>A . Some attempts to express these relationships may break down and generate logical paradoxes due to the inadequacies of global descriptions, but the underlying physics itself is paradox-free.
Some gravitational arguments also suggest that the gravitational potential between two points in space, measured along an agreed spatial route, may depend on the amount of time that a test object takes to traverse the route (dependence on initial velocity). For an object moving between two positions, route-dependence may apply not just to the spatial path but also to the spacetime path taken.
These arguments appear when we attempt to calculate the gravitomagnetic effects of the velocity of a body, and are more complicated.