If it does not work because it has been sitting unused for a very long time the grease gets sticky causing the problem. This is very common.

I would try to revive the shifter with any of the flush and re-lube techniques. It is free and only requires some effort.

If it is a shifter that has been in constant use then it may actually have broken part(s) . If that is the case no amount of flushing and re-lubing will help.

Good luck.