In this paper, an algorithm is presented to calculate the transition rates between adjacent mesoscopic subvolumes in the presence of flow and diffusion. These rates can be integrated in stochastic simulations of reaction-diffusion systems that follow a mesoscopic approach, i.e., that partition the environment into homogeneous subvolumes and apply the spatial stochastic simulation algorithm (spatial SSA). The rates are derived by integrating Fick's second law over a single subvolume in one dimension (1D), and are also shown to apply in three dimensions (3D). The proposed algorithm corrects the derived rates to ensure that they are physically meaningful and it is implemented in the AcCoRD simulator (Actor-based Communication via Reaction-Diffusion). Simulations using the proposed method are compared with a naive mesoscopic approach, microscopic simulations that track every molecule, and analytical results that are exact in 1D and an approximation in 3D. By choosing subvolumes that are sufficiently small, such that the Peclet number associated with a subvolume is sufficiently less than 2, the accuracy of the proposed method is comparable with the microscopic method, thus enabling the simulation of advection-reaction-diffusion systems with the spatial SSA.