Most of the models of alluvial fan and delta morphodynamics to date have used one of two assumptions concerning the upstream boundary condition: (1) a fixed sediment feed point at the vertex or (2) a feed point corresponding to a bedrock-alluvial transition that, e.g., migrates upstream under conditions of constant base level. Here, however, we present both experimental and numerical results pertaining to a new configuration, i.e., one in which the sediment feed point migrates downstream. The research is motivated by the operation of the tailings pond of a mine in which a delta forms. The mine operators must frequently move the outfalls of their slurry pipelines downstream to avoid their burial under sediment as the delta aggrades. This downstream migration results in a characteristic delta configuration. Under the right conditions, as the tailings delta progrades downstream, it does not have time to fill all the available space in the lateral direction, thus leaving unfilled open water on either side. The result is the formation of an elongated delta. Here we use both experiments and numerical modeling to characterize the control of the feed point migration rate on the lateral extent of the prograding deltas. We use these results to obtain a partial explanation of natural elongated deltas. In the case of such deltas, the downstream progradation of natural levees provides an analog to the downstream migration of the outfall of a slurry pipeline.