I've spent months trying to solve an issue with the rear shifting on my wife's bike. When downshifting from cog #3 to #2 on the rear cassette (inward from the 3rd-largest cog to 2nd-largest cog), the chain will temporarily be grabbed by the largest, innermost cog (#1) before dropping (unpredictably and jarringly) back out to the correct #2 cog.

Here is a video. I do two cycles in and out, so you can see the improper shift happen twice. You can also click the video’s settings->Playback Speed to run it at half-speed.
I think multiple factors are contributing to the problem, but one of the biggest seems to be “overshifting”, which is the term I’m using to describe the movement of the derailleur pulleys inboard past the targeted cog position before settling back out in line with the cog.

Here is a second video, with the chain off, which makes it easier to focus precisely on the derailleur movement. Again, you can slow it down to half-speed to make the “overshift” even more-obvious. I’m making a particular effort to push the lever the absolute minimum amount necessary to get it to click into the next indexed position. The final inboard shift doesn’t display the same amount of overshift because that’s where it’s stopped by the limit screw.
This seems to be a property inherent to indexed shifters, though I’m not sure if it’s an intended feature (to make downshifts happen faster/more-reliably?), or an undesirable side-effect. Anyone know? I can see it in this Park Tool video where on the downshifts, the bare cable advances slightly past the overlaid white lines before settling back (and in contrast, on upshifts it falls precisely to the final position, which is the same way my shifter works). Jump to 1:56; sorry, it seems like the embed doesn't allow me queue up the video to that point:
Unfortunately I don’t have other bikes/components to compare (my own bike also uses MicroShift shifters that I long-ago switched to friction-mode), and the full housing on my wife’s derailleur cable prevents me from visually comparing with the Park Tool video.

At the derailleur itself, on my wife’s bike, the pulley moves nearly one whole extra cog-spacing inboard (maybe 1-2 cog-thicknesses less) before settling back. This qualitatively seems like “a lot” to me, but I’ve never really looked at it before, so I don’t know if this is unusual. So that’s why I’m looking to the wisdom and experience of the group here.

My inclination is to replace the shifter and see if that’s sufficient to stop the problem. The current shifter is a Microshift SL-T09 9-speed thumb shifter. We like the simplicity and (theoretical!) durability of the shifter design (vs. trigger shifters, which don’t easily fit on her Koga Denham flat handlebar).

1. So the first obvious replacement would be Shimano Dura-Ace bar-end shifter grafted onto a Paul Thumbie. But if the Shimano shifter exhibits the same amount of overshift, I’d rather not waste all that money just to learn that. So: I would be very thankful if any Shimano bar-end owners here could report on what their “overshift” looks like!
2. A second option would be a simple replacement with a new Microshift SL-T09 (or even just its indexing parts). I’ve disassembled, cleaned, and reassembled the existing shifter, and didn’t find any obvious defects, but it’s certainly possible that the little balls and the indexing plate have worn to somehow cause “overshift” beyond spec. (the current shifter is only(?) three years old, with ~8000 miles on it, though we’re both high-frequency shifters).
3. A third option would be to switch to Microshift’s ADVENT line, with an 11-42t cassette.

The last option brings us to the 2nd-biggest contributing factor that I’ve identified: she’s currently running a Shimano 11-36t cassette with a Shimano XT RD-M772-SGS derailleur, which is technically specced (PDF) to handle only an 11-34t cassette. Though the Shimano Compatibility Chart seems to show it’s compatible with 11-36t, and everyone seems to say that Shimano’s specs can be loosened in practice anyway. And I don't recall this being a problem until (relatively) recently.

HOWEVER! If I run my own 11-34t cassette on her bike, the problem occurs much less frequently. So there must be something about that big 36t cog that, when combined with the overshift, makes it easier for it to grab the rising chain away from its neighboring cog when it isn’t supposed to.

My wife isn’t ready to give up her low gearing that comes with that 36t cog, so that’s why I wonder if switching to the ADVENT system that’s specifically designed for huge cogs (it can go up to 46t) would be the safest choice.

I'm a reasonably-experienced self-taught bike mechanic (I built up both my wife's and my own touring bikes that we've been living off of for the last 18 months), so I'm fairly confident that I've worked through the obvious possibilities, but I'm certainly open to any other ideas/solutions!

• Cable tension adjustment: shifting between all the other cogs never misfires (unless I adjust the tension such that the 3->2 shift doesn't overshift; in that case, all the other downshifts have trouble actually making it to their intended gear).
• Derailleur B-screw: I’ve adjusted it to both extremes (away from the spec’ed position I normally have it at), and it doesn’t seem to have much effect either way.
• Derailleur hanger: the derailleur hanger actually was significantly out-of-whack, but straightening it didn't seem to change anything in the shifting behavior (it was actually bent outward, which, combined with its lack-of-influence in the shifting issue, suggests to me that it had somehow been like that since I built up the bike; I never actually checked it after installing the hanger/derailleur onto the frame).
• Cable/housing: this is the one item I haven’t actually process-of-eliminated. I have removed, lubed, and reinserted the cable in the (full-length) housing, and it seems to slide freely. And the quick and reliable upshifting, when the derailleur spring pulls the cable back its way, suggests there isn’t a lot of resistance there. But, I haven’t actually replaced the ~3-year-old housing.

Again, thanks for any help and ideas. My wife learned to ride a bike when she was 35 years old, so it takes a lot less to shake her confidence than those of us with 2nd-nature biking skills, and I really want her to be free of the fear of these jarring shifts that happen at the worst possible time (dragging her 90lb. bike up an ever-steepening hill!)