The model developed in Part 1 (Lister 1994) for the solidification of hot fluid flowing in a thin buoyancy-driven layer between cold solid but freely deformable boundaries is extended to study the case of continual release of fluid. In this model lubrication theory was applied to reduce the equations of mass and heat conservation to a kinematic-wave equation and an advection-diffusion equation, which were coupled by the rate of solidification. The equations allow the source flux to be specified, and the cases of constant input and of flux proportional to a power of time are considered here. The structure of the flow differs significantly from the case of constant-volume release considered in Part 1. The advective resupply of heat prevents the flow from solidifying completely at the source and, if the initial fluid temperature is greater than the melting temperature of the solid, will in fact lead to rapid melting near the source. A perturbation expansion is used to describe the development of thermal boundary layers at the flow margins and the initial self-similar extension of the zone of melting. As the flow propagates beyond its thermal entry length, the fluid temperature falls to the liquidus value and melting gives way to solidification. At large times nearly all of the fluid supplied solidifies against the margins of the flow but, provided the source flux decreases less rapidly than t−½, sufficient reaches the nose of the flow that the flow continues to increase in length indefinitely. Analytic solutions are given for this long-time regime showing, for example, that the length increases asymptotically like t1/2 for constant-flux input. The theoretical solutions, which are calculated by a combination of analytic and numerical methods, may be used to describe the propagation of a dyke fed by a large body of magma through the Earth's lithosphere or the flow of lava down the flanks of a volcano during an extensive period of eruption.