Short scale doesn't make a difference as far as the tuners go, and tuners being locking or not is no indication of stability. There are lots of poorly-made locking tuners which wiggle more than high-end, well-made, non-locking tuners.
As long as you stick to the bigger brands, you should be okay. The Planet Waves locking & auto-trim tuners are absolutely rock-solid—sometimes a bit too solid—and Grover 18:1 Rotomatic, and Gotoh 301, 381 and 510 lines are all dependable, too.
The main thing to be aware of when trying to set up a short scale guitar to downtune often is how quickly the string tension can change—much more dramatically than when downtuning longer scale guitars—and the need to wrap and lock the string correctly, even if not using locking tuners. Short scale guitars shift in tuning so rapidly that if you start tuning down, by the time you've retuned the last string, the first string you retuned may have shifted a whole half step or more. So you've got to make sure the nut and bridge are as solid as the tuners, you should be prepared to tighten the truss rod perhaps a little more than you would with a regular guitar, and you need to be careful with your string selection. The overall set up of the guitar is far, far more important than any one piece of hardware.