Yes, cage edit is what I have been looking for. But it looks like a hassle to make it work in grasshopper, so in search of quick solutions, I stumbled over this, which is not exactly space deformation, but should work in my case.
My mesh has very many vertices, though. About 3000-4000. And the affected geometry is a brep. So until now nothing has happened after I set up the definition. How long should I leave it running before giving up @DanielPiker ?
There is also a spatial deform component in grasshopper that works by moving a set of reference points that affect geometry with a sort of force with a falloff function. Will try this soon.