Fundamental to interpreting the stratigraphic architecture within a basin is understanding the relationship between a basin's external conditions and its stratigraphic response. Here we present a model of fluvial basin filling that is based on two field observations. Firstly, natural fluvial systems commonly have an upstream region dominated by gravel transport and a downstream region dominated by sand transport with the transition between the two being relatively abrupt. Second, gravel bed and sand bed systems operate at nearly constant but different mean Shields stresses. Our model is based on a single, linear diffusion equation but is unique in that we use distinct transport (diffusion) coefficients for the two dominant fluvial regimes: a proximal gravel region and a distal sand region. This problem is complicated by the existence of two moving boundaries: the position of the distal fluvial toe and the position of the gravel-sand transition. We present a rigorous numerical treatment of both of these moving boundaries and verify our numerical formulation by comparing the model results to a semi-analytical solution technique. We use the model to examine the stratigraphic response to perturbations in four external boundary conditions: sediment supply, water supply, rate of subsidence and gravel fraction. The response is analysed in terms of the phase relation between forcing and the position of the gravel front, the position of the fluvial toe, proximal accumulation rate and distal accumulation rate. The model supports the results of earlier single-diffusion models suggesting that the form of the response is dependent on the period of the perturbation relative to the intrinsic basin response time. For forcing periods less than the intrinsic basin response time, basin response is nearly constant and independent of the forcing period, suggesting that the transport system controls basin response. For forcing periods greater than the intrinsic response time of the basin the response time of the basin increases directly with the forcing period, suggesting that the transport system plays no role in limiting basin response. For gravel-sand systems we show that the intrinsic response time is a function of the ratio of gravel to sand entering the basin. Forcing of the above external boundaries, both slowly and rapidly relative to the basin response time, produces both distal and proximal unconformities. We present a nondimensional 'unconformity number' that constrains the amplitude and period of forcing necessary to generate proximal unconformities.