I typed this out before I saw this had already been answered upthread, but I spent like five minutes on it getting the details right, so here goes:
In the original cast of Woods a body double was used. This was fudged for the filmed version of the show, where it's Bernadette Peters throughout, but onstage, the double (Maureen Davis, who played Sleeping Beauty) did the first half off the scene miming to Peters' offstage voice. I'm a little fuzzy on the mechanics of the swap, but if I recall correctly, as the Mysterious Man dies, Davis stepped behind a tree where Peters was waiting in an identical cloak to continue on the other side, and Davis made her exit via trapdoor. This same method was used in the broadway revival.
I'm less familiar with the ins and outs of the transformation In Beauty and the Beast but if I recall correctly: First off, the scene is helped with very dim lighting - I can't recall if a body double was used for the fight or not but I think there might have been. By the time of the "home" reprise, the Beast is covered with a cloak, allowing the wires to be attached and allowing him to remove the outer layer of his costume as well his. He is wearing a vastly-simplified, mask-like version of his make-up and wig, which has been removed during "The Mob Song" - the make-up allows for mouth movement but is not attached with any particular firmness. During the actual transformations, as the lights go nuts, all he has to remove his wig, gloves, and "mask" under the cover of flipping around, strobe lighting, and a quick blackout. I remember Jeff McCarthy telling a horror story of missing the wig one night and having to remove his horns once he landed.
Updated On: 1/21/17 at 08:30 PM