A quick search would show you that this has been discussed far too many times. Pick an oil you hold confidence in and run it. Many people use type F ATF (I do!), many use cheap 10w 30 (no friction modifiers!), many use synthetic MC specific oil (make sure it's wet clutch safe!), many use 80 or 90w gear oil. The transmission is obviously not too picky, but different oils can provide different feel in clutch and shifting.
To be accurate, the KDX doesn't use oil in its crankcase...it uses oil in its gearbox which is in-unit with the engine(crankcase). You will find no advantage in using anything other than what the factory recommends: 10w-30 or 10w-40 motorcycle oil. The factory also recommends changing it every 600 miles. I would change it AT LEAST that often.
Like I said above. Different oils will give you different feels of shifting and clutch action. It's completely personal preference, but there IS a difference. The transmission will handle just about any oil you like, so long as it is safe for the wet clutch setup. Quite true about the crankcase oil. 2 strokes don't have oil in the crankcase other than the 2t oil you mix in the gas.
And separate chambers is as real plus. Naturally you get this on a two stroke, but on four strokes it's a perk. My Buell is that way, most inline fours bikes are not.
The advantage is significant. Transmissions wear out oils much different then engines do. There are additives you can put in a transmission oil that make it last MUCH longer and work much better in gear intensive applications... but if these additives were to ever get in a combustion chamber they would leave deposits that would be really bad news. So they can't put them in motor oil, but can put them in gearboxes.
That's a complicated way of saying that if you can use transmission oil because your transmission oil is separate from your engine oil, you can get some specialize oils that will do much better then straight engine oil.
That being said, if you are changing oil after every 1000 miles, you won't wear *anything* out. So run what you want.
I like the ATF type F because it works well, and is nice and cheap, and is available at just about every wal mart and auto parts store I have been in.
Yeah, I got ya. A small difference can be an advantage though if it fits in with your riding style. That's all I'm getting at. I feel like my bike shifts smoother and has less clutch drag using Type F than 10w30. To me, that's an advantage. Could be placebo affect though, I dunno.
Anyway, like I said to the op, the transmission is pretty tolerant of whatever oil you like, just so it's safe for the wet clutch. If you run 10w30 don't get any fancy stuff that says "energy saving" on the bottle, it's likely to burn up your clutch. 10w30 or Type F are both inexpensive and work quite well. I can't see the point in spending over 10 bucks on a quart of gearbox oil to gain no advantage, like Mud said.