I tried liquify, but it does not do what I’m looking for. The “liquify” visual is not what I am after, even though it pushes pixels around it is not as general-purpose as I’d like.
I found a way to do something similar to what I wanted with painting with a deformation brush using a selection mask (open the GIF in a new tab to actually see the details):
The main drawback is that the texture mask is fixed (instead of being a stencil moving with the brush), but that’s acceptable. AFAIK having a brush texture mask is only available for brushes created with the Pixel engine, i.e. not ones that push pixels around.