I seem to remember that the Falcons do come apart like desribed, but require a lot more effort. I'm a fan of just trashing them and replacing with a SRAM MRX Comp. They're less than $10, include the cable, work better, and are easier to service.
Ah, yes, it's that "require a lot more effort" part that has me worried. The Falcon shifter doesn't look that robust. I fear I would break it even in taking it apart correctly.
Thanks for the heads-up on the MRX Comp. I hadn't been aware of those. Having an easy replacement makes the risk of breaking the shifter much easier to take. Jenson has them for less than $9.00. I might go ahead and order one.
I've just done this.
There's a curved cover that snaps in between the cable adjuster and the rotating part of the shifter.
Pry off the end near the cable adjuster. Then comes the difficult part. There's a plastic post on this cover that slides into the main body that will make it difficult to pry off the other end of the cover. Alternately work the cover up along the axis of the post and try to pry the cover from the main body at the grip. Takes a bit to pry to cover off.
Rotate the shifter to let out the most cable. There's probably a little round cap covering the access hole on the bottom of the cover (the one I had had this missing) Pry the cap off. Push the cable back into the shifter. You should get a loop form where the large cover was. Pull the cable out from there. then push it back into the main body of the shifter until you see the head at the access hole. Probably will need something pointy to help the head pop out. Pull out the cable. It's not much different to changing the cable on a SRAM gripshift.