Blocks that are attached to other blocks (such as buttons) tend to be movable with Create's movable structures (such as, in my case, an elevator). This doesn't appear to be the case with signs, even when using superglue. When supported, I hope it's possible for the signs to continue displaying the text on them while moving.

Also according to the v0.3.1b changelog, signs can now be moved.
Will close the issue. Thank you 💚