Hi bluesman0007,
Typically what happens when you do that is a "stick-slip" motion, where the load accelerates out of control until the stretched shock cord or spring relaxes, then the load stops, then the shock cord or spring stretches again, then the load moves again, etc. Theoretically one would like to have an infinitely rigid (in tension) cable to avoid this erratic motion. Often, inserting a spring or shock cord will make matters worse.
The wire rope is a spring in itself. For instance, a 1/2 inch cable, 6 X 19 IWRC Carbon construction, has an approximate spring rate of 0.0001 inches stretch per 100 loaded inches of cable per pound load. Fortunately as you near the stop the loaded inches of cable reduces so the spring rate gets stiffer.
Try to run the rig with as low a line speed as practical. In addition you can run a tag line around a viscous damped capstan to steady the motion. Dashpot bumpers at the end of the pull are also helpful.
Rig a multipart line block and tackle as this will give a equivalently higher stiffness and at the same time a lower load speed.