I'm wondering how the option "Texture Mesh" in the "Texture Space Panel" of the "Object Data Properties Tab" is working:
File is here.
According to this post here, it should work with a duplicated mesh where you can move the vertices around in order to influence the texture mapping on the original object. I can't make Blender working like this. Maybe, it's a remnant from "Blender Internal Render Engine"?