What NJ said - only I disagree with the last part. If it's that dry it's REAL dry. Spray until it's overflowing, give it time to run down the inside of the cable - then hit it again. Repeat a few times. Put a rag wherever it's flowing out. You won't hurt anything and you'll be amazed at how much some good cable lube affects the pull. JMHO. YMMV.Go to the bike store and buy a can of Cable Lube. You can ask which is the best one. It comes with 'straws' that direct the spray where you want it. Weasel the straw into the space between the steel clutch cable and the outer casing. Hold a rag there in case it oversprays. Give it a shot and check if the stiffness is getting better.
Don't use too much, it comes out the other end and makes a mess.