If you're seeing the ceiling insulation slowly losing its shape under the joists, that temporary push-up method won't solve the underlying issue. You want that material fully compressed and secured right from the start, not just forced into place for a quick look. When installing fiberglass batt, make sure you are using the right amount of force to get it seated tightly within the framing before you move on to the next section. If it's loose and simply resting on the joists, it will naturally shift over time, expecially in fluctuating temperatures. It needs to be firmly wedged in.
For the drywall phase, if you manage to get the batts fully seated and snug, the chance of seeing movement is very low. Just make absolutely certain all the edges and seams are taped and fastened correctly. Once that bedding is solid, the drywall layer will just cover it up and hold it in place.