The intrusion of a polydisperse suspension of particles over a horizontal, rigid boundary is investigated theoretically using both an integral ('box') model and the shallow-water equations. The flow is driven by the horizontal pressure gradient associated with the density difference between the intrusion and the surrounding fluid, which is progressively diminished as suspended particles sediment from the flow to the underlying boundary. Each class of particles in a polydisperse suspension has a different settling velocity. The effects of both a discrete and continuous distribution of settling velocities on the propagation of the current are analysed and the results are compared in detail with results obtained by treating the suspension as monodisperse with an average settling velocity. For both models we demonstrate that in many regimes it is insufficient to deduce the behaviour of the suspension from this average, but rather one can characterize the flow using the variance of the settling velocity distribution as well. The shallow-water equations are studied analytically using a novel asymptotic technique, which obviates the need for numerical integration of the governing equations. For a bidisperse suspension we explicitly calculate the flow speed, runout length and the distribution of the deposit, to reveal how the flow naturally leads to a vertical and streamwise segregation of particles even from an initially well-mixed suspension. The asymptotic results are confirmed by comparison with numerical integration of the shallow-water equations and the predictions of this study are discussed in the light of recent experimental results and field observations.