A fluid-handling cassette comprising a plurality of diaphragm valves and pumps is configured to have its actuation ports located along a thin or narrow edge of the cassette. Actuation channels within the cassette lead from the actuation ports to actuation chambers of the valves and pumps in a space between plates that comprise the cassette. The individual plates have a nominal thickness that is sufficient to provide a rigid ceiling for the actuation channels, but sufficiently thin to minimize the overall thickness of the cassette. The cassette can be plugged into or unplugged from an actuation receptacle or a manifold by its narrow edge. A plurality of such cassettes can be stacked together or spaced apart from each other to form a cassette assembly, providing for a convenient way to install and remove the cassette assembly from its actuation receptacle. The arrangement allows for an improved way of connecting a complex cassette assembly to its associated pressure distribution manifold without the use of a plurality of flexible connecting tubes between the two.