Actually, this is just a special case of the more general "space tether" problem, in which the entire system moves at the orbital rate of the center of mass. Things below the CM travel slower than they should, and things above the CM travel faster than they should.
The whole key to the space elevator is to maintain the Center of Mass of the entire system at GEO. If you lower something down, something else has to go up, in order to keep the whole system at GEO.
If you drop something off the elevator, you've essentially moved the CM of the system down. The bigger the thing you drop off, the farther down the CM drops. The new CM will be travelling more slowly than it should, so I believe the whole tether will begin to sag, until the CM can be brought back to GEO.
Likewise, anything latching on to the tether will raise the CM, and the tension on the tether should increase some, until the counterweight can be moved to compensate.
I think in practice the mass of the tether will be vastly greater than anything being dropped off of it, so the net effect may not be that large.
The in-plane location of the CM would be somewhat controlled by adjusting the CM. Out-of-plane position would require the use of thrusters.
The question of lunisolar gravitational perturbations (Don Joe's "tidal forces") are serious -- if nothing else, they represent a significant periodic tension variation on the tether.
It will stabilize wherever the base is anchored even as far north as the continental US. You can be pretty far off the equator and still get the thing to work.