I have seen the coral move itself at least twice. Each time it only moved a few millimeters. Just before moving the coral changes from brown to green and partially retracted its tentacles. Then, all at once it pushes itself by swelling the tissue that is in contact with the sand bed. It then extended its tentacles and turned brown again. It moves itself just a little bit and then goes back to being its normal sedentary self.
This time I'm going to watch it closely and document its progress as it moves. Here is a photo of its current location.