Some guitars are less susceptible to this than others. My Mugen has a rather flexible table and is a pain for this purpose. Old EKOs are built like cricket bats and as stable as all hell (but don't sound do good). THe Landola 12 strings have a gadget like an upside down truss rod from the inside of the saddle to the tailblock which makes them very stable but a bit thinner sounding than many 12s.
Locking tuners, either of the type already described, or locking nuts (ie a nut with a bar over it held down by little bolts to clamp the strings at what is analogous to the zero fret) are not likely to solve this problem.
However, the problems of an acoustic in this situation are nothing to that of a plank with a fully floating floyd-rose trem, which wobbles like a jelly on a plate when tuning.