A cleaning or care device for the cleaning or care of medical or dental instruments, having a cleaning or care chamber in which at least two connections for the instruments to be cleaned are provided, and a supply device for supplying the at least two connections with a cleaning or care agent. The at least two connections and the supply device can be moved relative to one another in such a way that one of the connections can be connected to the supply device and supplied with a cleaning or care agent, while another of the connections is not connected to the supply device and cannot be supplied with a cleaning or care agent. A corresponding method for the cleaning or care of medical or dental instruments with such a cleaning or care device is also described.