A drag and/or dip finishing machine has at least one workpiece holder for the releasable fastening of the workpieces to be machined and a container, disposed beneath the workpiece holder, for accommodating the grinding and/or polishing granules, wherein the workpiece holder and the container are movable relative to each other in that at least the container is rotationally driven. The container has a chamber, which extends around the whole of the periphery thereof and is fluidically connected to the inside, accommodating the grinding and/or polishing granules, of the container, at least on a lower portion thereof, the chamber being equipped with a liquid discharge unit in order to be able to discharge the machining agent from the chamber in the event of an excess of machining agent in the container.