(gets out soapbox and ultrasonic cleaner)

What the OP describes is a classic case of an older Shimano shifter gumming up as the grease ages and hardens, as correctly diagnosed by several replies. WD-40, White Lightning Finish Line Clean Streak, PB Blaster, and other solvents can be effective in temporarily fixing the problem. But over time these solvents will evaporate, the grease will re-solidify, and the problem will recur. If it's your own bike, periodically shooting the shifter with solvent should work to keep it working. But for a bike that's going to others, I don't feel all that great about selling or giving away what is in effect a ticking time bomb in terms of shifting.

The only way to permanently fix the problem is to (a) remove the shifter, optionally hurl it against a wall*, and replace it with one that isn't as susceptible to the problem. Or (b) you can remove the shifter and get the gunky grease completely out of it with an ultrasonic cleaner. Which takes time. Lots of time.

Here is my definitive posting on the process:
De-Gunking Balky Shifters - For the Long Haul

(puts away soapbox and ultrasonic cleaner)

* Not recommended, as you might hit someone else or damage something valuable. Plus the shifter might not be able to be cleaned ultrasonically after this treatment.