I think this will be hard to explain but here is what I am trying to do.
Suppose we have 2 solid cube on top of each other (larger cube on the bottom and smaller cube on the top). I mesh the top cube with fine mesh (quad mesh surface and then extrude).
On the bottom cube (adjacent surface to the top cube) I would like to imprint the mesh from the top surface onto bottom and use the same boundary and mesh the remaining surface such that the imprint mesh stays same.
If you are working with HexMesh, then you can still do the Edit > Solid > Imprint step, but then you would want to mesh the outer area of the cut face of the with quads, skin the solid mesh on the face of the inner area to get the quads there. Then you can Sweep > Element > Extrude the quads to get HexMesh the rest of the larger cube geometry.
If you are working with HexMesh, then you can still do the Edit > Solid > Imprint step, but then you would want to mesh the outer area of the cut face of the with quads, skin the solid mesh on the face of the inner area to get the quads there. Then you can Sweep > Element > Extrude the quads to get HexMesh the rest of the larger cube geometry.