The problem comes in the following case, when the application of a dress on a rigged mesh is moving. The dress is applied using the surface deformer which is based on the character’s body. The shift is visible both in the editor and when rendering an animation.
We see that the surface deformer uses the previous version (frame -1) of the body.

The solution is simple, simply reverse the order of objects in the hierarchy by placing the rigged body before the dress applied to the surface deformer.

The evaluation order is the following:
1) Depending on the priority group.
The hypernurb, the skin and deformer belong to the group of generators. So they have the same group of priority.
2) In the same group, depending on the specific priorities.
The skin has a priority of -1 and will be the first of the group to be evaluated.
3) according to their positions in the hierarchy.
The hypernurb on the body is processed first, then the surface deformer on the dress.