Experiments are reported on the sustained release of saline and particle-laden fluid into a long, but relatively narrow, flume, filled with fresh water. The dense fluid rapidly spread across the flume and flowed away from the source: the motion was then essentially two-dimensional. In the absence of a background flow in the flume, the motion was symmetric, away from the source. However, in the presence of a background flow the upstream speed of propagation was slowed and the downstream speed was increased. Measurements of this motion are reported and, when the excess density was due to the presence of suspended sediment, the distribution of the deposited particles was also determined. Alongside this experimental programme, new theoretical models of the motion were developed. These were based upon multi-layered depth-averaged shallow-water equations, in which the interfacial drag and mixing processes were explicitly modelled. While the early stages of the motion are independent of these interfacial phenomena to leading order, they play an increasingly important dynamical role as the the flow is slowed, or even arrested. In addition a new integral model is proposed. This does not resolve the interior dynamics of the flow, but may be readily integrated and obviates the need for more lengthy numerical calculations. It is shown that the predictions from both the shallow-layer and integral models are in close agreement with the experimental observations.