This paper presents a Lagrangian model for particle transport driven by a two-dimensional (2D) shallow water model, assuming that the particles have negligible mass and volume, are located at the free surface, and without interactions between them. Particle motion is based on advection and turbulent diffusion, which is added using a random-walk model. The equations for particle advective transport are solved using the flow velocity provided by a 2D shallow water solver and an online first-order Euler method, an online fourth-order Runge-Kutta method and an offline fourth-order Runge-Kutta method. The primary objective of this work is to present the capabilities of the new Lagrangian particle transport (LPT) model, while also providing an analysis of the accuracy and computational efficiency of the numerical schemes and their implementation for particle transport. To verify the accuracy and computational cost, several test cases inspired by laboratory set-ups are simulated. In this analysis, the Euler online method provides the best compromise between accuracy and computational efficiency. Finally, a localized precipitation event in the Arnás catchment is simulated to test the model's capability to represent particle transport in overland flow over irregular topography.