A lot of auto companies used to use Dextron III several years ago because it was sort of the defacto industry standard -- the best of the best for a long time. These days, however, there have been recent advances in transmission fluids and some mfgrs have broken ranks and gone with like ATF +4 and some others which, if you read up on them are more efficient on shear pressures. It's entirely possible Mitsubishi fluid is re-packaged Dextron III, but made by who? And maybe that's important. Transmissions are finicky to the type of ATF ya use. Maybe you can see if the parts guy at Mitsubishi can give you a deal on their stuff, and ask them or the service manager what their experience is.
I've used Dextron III in lots of applications (GM vehicles) for many years that originally called for DII, and never had a problem. It supersedes DII. However, I have had issues using different mfgrs of Dextron III, only once though. I have a older GMC pickup (89) 4x4 that calls for Dextron III. I replaced the fluid with Valvoline DIII, which I've used in other vehicles for many years w/o any problems. After about 100 miles the tranny started freaking out, shifting oddly, wouldn't go into overdrive and was shuddering badly. I was planning to get it rebuilt as it is a high-mileage unit. A couple days before I was getting ready to pull the trigger on a $1,000 tranny, I did some internet searching and happened upon a post that described how some people have had such shudder problems go away using Penzoil Dextron III as supposedly it contains more anti-shuddering agents in it. Indeed they even include it on the Penzoil ATF label. I was skeptical, but figured it was worth a shot. After about 50 miles all the problems I was experiencing completely went away and I was flabergasted. That was about 6 years ago so I'm glad I did it! I only put about 4000 miles on it a year so for what I use it for, not needing a rebuilt tranny was really great. But still, I was surprised that it worked.
In other instances, some older Chrysler vehicles for may years, like Jeeps and other Chryslers in the 90's, per their manual, you could use Chrysler ATF, or use generic Dextron III and add an approved additive package, which was basically a pint of special anti shuddering agents (and who knows what other modifiers were in there), that modified the ATF to meet Chrysler specs. Done it many times, and know a lot of other guys that go that route. Sometimes guy will just get the Chrysler stuff as it already had the additive in there. If you get the mechanics' price on parts from Chrysler, their stuff isn't any more expensive than parts store stuff. In fact, the last ATF+4 I bought, Chrysler was like $2 per quart cheaper than the parts stores!!
As for getting it all out, what I do is change my vehicle's fluid, but don't start the vehicle. Add about 2 extra quarts, then pull one of the cooling lines off at the radiator up front. Slip a hose over the line ...and another hose into the threaded end (use a barbed fitting as necessary), and use clear vinyl hose if you can. Then stick the hoses into a 3-5 gallon bucket underneath the radiator and have a helper start the vehicle. As soon as it starts, fluid will come gushing out one of the hoses into the bucket. (The reason to use two hoses is because it's hard to predict which hose the fluid will come out of unless you are familiar with the transmission fluid path). Anyway, watch very carefully the fluid coming out. Use a strong flashlight if it's dark. After about 12 seconds or so you'll see the fluid become clear indicating that you've basically emptied all the old fluid out of the torque converter and replaced it with the clean stuff that was in your transmission pan. Then signal your helper to shut the vehicle off and top it off as necessary. You'll probably get out about 1.5 - 2 gallons of fluid, it comes out very fast. If it takes longer than you think shut off the vehicle and add a couple quarts so you don't run the tranny dry, but if you overfill it a little, you shouldn't run it dry. There .... you've just flushed the fluid out of the torque converter. You won't starve the tranny because the cooling lines are the 'return' lines that pump back to the transmission pan. This doesn't work, obviously, on closed transmission systems w/o coolant lines.
The other thing I usually do during a transmission fluid change, is take a fine-threaded 1/2" nut from the hardware store and cut it short with a hacksaw, say, about 1/4" thick. Then braze it to the outside of the tranny pan and put a bolt in there with a gasket on it. They make kits for this, but they cost a lot more and I never have them on hand. One day I finally had enough and went and bought several bolts and nuts and cut them down and put them in a bag so when I need to do a tranny fluid change, I have it ready to go. Saves a lot of messing around.