A method for conveying viscous material out of a barrel-like container open towards the top and having a bottom and a circumferential wall extending from bottom to top, uses a device having a follower plate for closing the container. The plate has a circumferential wiper ring having a radially outwardly protruding contact portion for contacting an inner container wall face facing the plate. A variable pressing force can be applied to press the contact portion against the inner face. The plate, following a decreasing material level and resting on the material surface, is moved bottomward, and, after the container is emptied, is moved away from the bottom, and the contact portion is pressed sealingly against the inner face while the plate moves bottomward. The sealing effect between the wiper ring and the container wall is suspended during plate movement away from the bottom by reducing the pressing force.