gwoodsii

I know that traveling up a space elevator slows the earth slightly and traveling down it speeds it up slightly. I also know that this is a miniscule amount. I am curious though if this effect continues past the geosynchronous orbit point if the load were to continue on the cable towards the counter weight? Also would any type of engine exerting a force on any point of the elevator also change the speed of rotation or would it simply destroy the elevator?