We employ shallow water analysis to model the flow of particle-driven gravity currents above a horizontal boundary. While there exist similarity solutions for the propagation of a homogeneous gravity current, in which the density difference between the current and ambient is constant, there are no such similarity solutions for particle-driven currents. However, because the settling velocity of the particles is often much less than the initial velocity of propagation of these currents, we can develop an asymptotic series to obtain the deviations from the similarity solutions for homogeneous currents which describe particle-driven currents. The asymptotic results render significant insight into the dynamics of these flows and their domain of validity is determined by comparison with numerical integration of the governing equations and also with experimental measurements. An often used simplification of the governing equations leads to `box' models wherein horizontal variations within the flow are neglected. We show how to derive these models rigorously by taking horizontal averages of the governing equations. The asymptotic series are then used to explain the origin of the scaling of these `box' models and to assess their accuracy.