When you say "the component moves but the chain remains on the same gear," are you saying that the front derailleur moves normally when you operate the shifter? I'm trying to imagine how this would even be possible (not doubting your words, though). A front derailleur is such a simple mechanism. Sure, it's normal to have rough shifting or even noisy shifting if the derailleur is out of adjustment; however, as long as the derailleur is able to move across the chainrings, it should still shift the chain.

How long has this been an issue? Did the problem slowly develop, or did it happen suddenly? Do you believe a fall or impact may have caused this?