Nice job, PlatyPius. I take a different approach, admittedly less thorough but far less time consuming.

I call the phenomenon "dead shifter syndrome." You see a lot of bikes with this problem when the cold weather hits, that old factory lube is especially problematic in the cold. I usually can bring them back to life with a can of WD-40 equipped with the little red straw that comes with it. Install the straw, and with the shifters still on the bike, find a place you can get inside the shifter pod with the straw. Often, moving the shift lever will create enough of an opening, all you have to do is get inside with the straw somewhere.

Then, let 'er rip with the WD-40. Flush that sucker until you've got WD-40 coming out at every seam and gap on the shifter pod. Take a moment to work the shift levers, both upshifting and downshifting. Repeat as necessary. They'll more often than not come back to life within 30 seconds.

The WD-40 is mostly solvent, and will dillute and spread the thinned factory lube around enough that you're "re-lubing" enough as it is. I've got shifters that had this problem years ago, flushed them out like this, and they're still going strong. YMMV-