Abstract: We consider the approximation by multidimensional finite volume schemes of the transport of an initial measure by a Lipschitz flow. We first consider a scheme defined via characteristics, and we prove the convergence to the continuous solution, as the time-step and the ratio of the space step to the time-step tend to zero. We then consider a second finite volume scheme, obtained from the first one by addition of some uniform numerical viscosity. We prove that this scheme converges to the continuous solution, as the space step tends to zero whereas the ratio of the space step to the time-step remains bounded by below and by above, and under assumption of uniform regularity of the mesh. This is obtained via an improved discrete Sobolev inequality and a sharp weak BV estimate, under some additional assumptions on the transport flow. Examples show the optimality of these assumptions.