Move as in as, blow back and forth in a wave, or travel across your tank? Not coral "picks up and moves" like an anemone would, but brains and plates definitely wander around the sand.till they like the spot. All be it cm by cm...
The main grouping of coral I haven't heard of actually "moving" are stony coral. They basically lay out a skeleton and grow out, but don't technically move the way a mushroom or button polyp would. Most soft coral will move to try and get their preferred lighting and flow. Mushrooms will detach and re-attach in new locations, sometimes leaving little split pieces behind that grow into new ones.