The aim of this paper is to present a finite volume kinetic method to compute the transport of a passive pollutant by a flow modelled by the shallow water equations using a new time discretization that allows large time steps for the pollutant computation. For the hydrodynamic part the kinetic solver ensures - even in the case of a non flat bottom - the preservation of the steady state of a lake at rest, the nonnegativity of the water height and the existence of an entropy inequality. On an other hand the transport computation ensures the conservation of pollutant mass, a nonnegativ- ity property and a maximum principle for the concentration of pollutant and the preservation of discrety teady states associated with the lake at rest equilibrium. The interest of the developped method is to preserve these theoretical properties with a scheme that allows to disconnect the hydrodynamic time step - related to a classical CFL condition - and the transport one - related to a new CFL condition - and further the hydrodynamic calculation and the transport one. The CPU time is very reduced and we can easily solve different transport problems with the same hydrodynamic solution without large storage. Moreover the numerical results exhibit a better accuracy than with a classical method especially when using 1D or 2D regular grids.