I have a knee injury that is making biking difficult. I usually love climbing in the mountains, but it's a challenge now. I have a bike with 7900 10 speed Dura Ace, and another with 10 speed Tiagra. The 7900 bike has a 28T biggest cog (I think the same on Tiagra, but I'm not sure and might be wrong). I'd like to go up to at least a 32T cassette. I've done some searching and I've read that it's possible to drop in a 9 speed Shimano mountain derailleur. Is that true? Is it ANY shimano nine speed mtn rear derailleur? I've read it should work with the existing STI shifters.

Here's where things get fuzzy for me. I've read that in addition to that 9 speed mtn derailleur that I'm supposed to use a 10 speed road cassette. That seems to defeat the point a little bit, but I know there are Shimano road cassettes that go up to 32T, so I imagine that should be fine. Is that true that I need to use a 10 speed road cassette?

Thanks.

Yes any nine speed shimano derailleur that can handle a 32t, not sure what you mean by 10 speed road cassette defeating the purpose, the rear cog spacing is the same for mtb v road so any cassette so long as it matches your shifters ie 10 with a 10speed. Are you planning on swapping both bikes over? What version of Tiagra are you running the latest 4700 will not work with a mtb 9 speed derailleur as they have changed the pull ratio, depending on what version and dérailleur you have it may be as simple as a cassette change. There is also the option of running a compact crankset.

Derailers as a rule aren't that picky about the number of speeds they are used for. Shimano used to be very interchangeable, as most used the same actuation ratio. Not too sure about the "rules" for 10-speed road/mtb mixes, I think Shimano did something unfriendly with the introduction of Dyna-Sys.

Any 9-speed Shimano MTB rear derailleur will index properly with any Shimano 10-speed road shifter (except Tiagra 4700) and 10-speed cassette. The incompatibility comes with 10-speed MTB derailleurs and 10-speed MTB shifters and 11-speed anything.

I ride a heavy Walmart bike and it seems to me that even though it is a 29'er, the gearing if for maybe a 26" wheel and the low gears are not low enough, at least for me and my area. It has a comfortable ride but the slightest of hills make you work. This is a front suspension only "mtb" but I could never see it off road, unless maybe a smooth MUP.
I still like it for the $129 I got in it. I did reassemble it myself and lube everything. It's a Genesis GS29

Correct, you can use just about any 9 speed Shimano MTB RD and it will index perfectly with your 10-speed road shifters; they both use the Shimano's SIS standard.

The reason you have to use a 9 speed MTB RD is that none of Shimano's 10 speed road RDs fit a 32t or larger cog (except Tiagra 4700, which isn't SIS), and Shimano's 10 speed MTB RDs use a different indexing standard, "DynaSys".

You can use a short-cage (SS) or mid-cage (GS) MTB RD with an 11-32t cassette and a double crankset. The long-cage MTB RDs are intended for use with triple cranksets.

You can use any Shimano (or SRAM) 10-speed cassette; even though Shimano's indexing standard (cable pull) is different between 10-speed road and MTB, the cassette spacing is the same for both (and SRAM 10s cassettes). Personally, I use Shimano XT 10-speed cassettes because I get them reasonably cheap from bike-discount.de in Germany.

Lastly, you'll need a new chain to fit the larger cassette. Personally, I hate Shimano's chains because they don't have a re-usable quick-link. SRAM's 10 speed chains include a quick-link, but they claim it's for one-time use only (others have successfully re-used it, apparently). Personally, I use KMC or Wippermann/Connex 10-speed chains because they include a re-usable quicklink. The Connex link is the easiest to use, and it can be used on any chain.

Thanks so much for the replies everyone. To answer a few questions and follow up:

I'm not sure which version of Tiagra my bike is, but I think it's one of the early versions of the 10 speed for Tiagra, so maybe 4700? However, it looks like this bike might already be okay as it is reported online to have a 34T cassette already. I know, I should know this. But I'm at work and I only rode this bike as a commuter when my knee was healthy, so it never really mattered and I didn't usually use that low gear. Someday I'm thinking of switching this to bar end shifters for cost and maintainability down the road, but probably only when the STIs fail or when I can't find bar ends that'll handle 10 speed well anymore. Who knows how many years down the road that'll be.

I would like to convert the 7900 to a wider range cassette. (Well, I'd prefer my knee to get better, but failed surgeries and stumped doctors indicate that's not likely). So it sounds like I can grab any Shimano 9 speed mountain derailleur and any 10 speed cassette, mtn or road. I was initially hesitant about needing a road cassette because I thought they only went up to 32T, which is probably what I'd target, but it would be nice to have a 34T or 36T option. I already have compact cranksets on both bikes, so I'll probably be fine with the short cage version (SS). So, in summary, I'll need a new chain, cassette, and derailleur. Thanks again.

And in doing some research it looks like I have Tiagra 4600 (it has the window with the shift position indicator). So even if I wanted to change that to a 9 speed mtn derailleur, I could?

Junior racing cassetes start out with a bigger high gear cog , a 14 or 16 rather than the 11 & 12 t so common these days.

of course you can always set the high gear stroke-limit-screw and block out the smallest cog ..

Bar end shifters are believed to be more durable than STIs because they have fewer moving parts (and are almost completely metal). Shimano still sells BS79 10-speed bar-ends (and 8 and 9 speed bar-ends are also still available). I think you'll be able to find them for a long time if you wish.

I don't think Shimano offered any of their 9-speed MTB RDs in short-cage (SS) versions; I think they only came in mid-cage (GS) and long-cage (SGS) versions, for double and triple cranksets, respectively.

Personally, I really like the 9-speed XT RDs (XT is roughly Ultegra-level in the MTB lineup).
RD-M770 is "normal" cable routing and Rapid Rise (reverse order shifting)
RD-M771 is "normal" cable routing and shift order
RD-M772 has "shadow" cable routing and "normal" shift order

Or, you could get an XTR RD (XTR is roughly Dura-Ace level)
RD-M970 is "normal" cable routing and Rapid Rise (reverse order shifting)
RD-M971 is "normal" cable routing and shift order
RD-M972 has "shadow" cable routing and "normal" shift order

Shadow cable routing means the cable doesn't have a big loop from the cable stop to the RD, because where the cable enters the RD is up on the top and angled forward.
Shadow also has more of the derailleur upper parallelogram underneath the hanger and cassette, so it's more tucked out of the way.

However, Shadow RDs DO NOT have a cable adjustment barrel (because MTB shifters do). If you use a Shadow RD with a road shifter, you should add an adjustment barrel to one of the cable stops, or use an inline adjuster on one of the cable loops.

The Tiagra mid cage (GS) rear derailleur is rated for a 30t max cog, but it sounds like users have success with 11-32t cassettes. For a 34t or larger cog, a MTB derailleur would work better.

This is all really helpful info, thanks so much!