This is for the Shimano A20 "rapidfire" shifter switch (on the handlebars) but a similar situation may exist for other brands/models. The shifter consists of two levers and an enclosed ratchet mechanism with pawls. If pushing on the lever fails to move the idler wheels of the derailleur (on the chain), then a likely cause is lack of lubrication. If the old lubricant gets hard, the pawls will stick.
So remove the switch and then remove a cover. Push-pull the pawl that's giving trouble with a small screwdriver and watch what happens. The pawls have tiny springs that push them into contact with a lugged wheel. If a pawl fails to maintain contact with the lugged (or notched) wheel, then, if the spring isn't broken, it can be fixed by lubing the pivot point of the pawl. Work the pawl back and forth with a small screwdriver, etc. right after applying a drop or two of lube oil. When the spring freely moves the pawl into contact with the lugged wheel, it's fixed.