What is the fastest way to do a cut extrude / inward extrude? Let's say after a knife project I have a nice region that I want to extrude. Extruding outwards along the normal works fine. Extruding inwards along -Z however does not work the way I want it to.
Starting geometry with selection to be extruded inward:


Dragging this down generates correct faces in the interior but also zero volume regions at the exterior:


Instead, I am looking for a final mesh without these exterior slivers. Similar to ZBrush's QMesh or Solidworks Extrude Cut:


This is obviously a simplified example where I can delete offending faces one at a time. What is the best workflow to either avoid these faces or quickly select them all?
Extrude the face you don't want to move then RMB to leave the duplicated edges in place.  Now select the face you want to move and you get the behaviour you want, except there is an edge loop that has been left behind.  Use Alt click to convert the selection to vertices, then Shift + Alt + RMB click the edge loop.  Translate the selection and you're done.