First things first and that's compatibility of the lever and RD. Sram makes twist-grips for Sram RD's and Shimano RDs, but they're not the same. The ones for Sram usually have numbers like 7.0, 9.0, X.0 etc., the ones for Shimano have names like Comp, Rocket, etc. So step one is checking that you have a shifter that matches the RD.

The next stem is to check for smooth cable movement. Start by shifting to high gear and, while pedaling, pulling the bare wire away from the frame at the down tube as if drawing a bow. Watch the action of RD and see if it moves smoothly. Shift to low gear this way, and hold it there, while you use the twist grip to shift one gear at a time. You should feel it drawing your hand back smoothly with each click. If you feel the cable isn't moving in either of the above tests, you either have worn or sticky housings, or maybe the wire it trapped under something. Now it's just a bit of detective work.