Anomalous diffusion has recently turned out to be almost ubiquitous in transport problems. When the physical properties of the medium where the transport process takes place are stationary and constant at each spatial location, anomalous transport has been successfully analysed within the Continuous Time Random Walk (CTRW) model. In this paper, within a Monte Carlo approach to CTRW, we focus on the particle transport through two regions characterized by different physical properties, in presence of an external driving action constituted by an additional advective field, modelled within both the Galilei invariant and Galilei variant schemes. Particular attention is paid to the interplay between the distributions of space and time across the discontinuity. The resident concentration and the flux of the particles are finally evaluated and it is shown that at the interface between the two regions the flux is continuous as required by mass conservation, while the concentration may reveal a neat discontinuity. This result could open the route to the Monte Carlo investigation of the effectiveness of a physical discontinuity acting as a filter on particle concentration.