If I try to use the Dimension Shift ability while affected by Freeze in Place, the champion just vanishes from the board. When I tried the ability Swap with Minotaur Battlemage (in a different game) while under the spell, the swap animation happened and then the champions returned to their original spaces, with Battlemage losing the 2 AP in the process. So yeah, the vanishing doesn't happen on all relocation effects.
Just tested again, with both the Minotaur Culler near the Door and a Minotaur Sentry. With the Culler, I targeted the space diagonally up from the Door; it simply triggered the purple relocation effect, lost 2 AP and stayed in place. With the Sentry I targeted the little white space highlighted in the image. It vanished.
