Movement of a piece along the axis of rotation (the Z axis).
There are two YouTube videos showing this is below. The first one is on a Lindow-White Rose Engine (which was purchased).
Ornamental Turning with the Lindow White Rose Engine Side Pumping
The second one shows pumping on an MDF Rose Engine (which Pat Miller made). This is a very innovative approach that Pat made, using the rosette designed to move the object in a rocking motion, and converting motion that into a pumping action.
Pump Cutting on the MDF Rose Engine