The flow characterization of blood through healthy and diseased flow geometries is of interest to researchers and clinicians alike, as it may allow for early detection, and monitoring, of cardiovascular disease. In this paper, we use a numerically efficient particle-based flow model called multiparticle collision dynamics (MPC for short) to study the effect of compressibility and slip of flow of a Newtonian fluid through a cylinder with a local constriction. We use a cumulative averaging method to compare our MPC results to the finite-element solution of the incompressible no-slip Navier-Stokes equations in the same geometry. We concentrate on low Reynolds number flows [ Re∈(4,30)Re∈(4,30) ] and quantify important differences observed between the MPC results and the Navier-Stokes solution in constricted geometries. In particular, our results show that upstream recirculating zones can form with the inclusion of slip and compressibility, which are not observed in the flow of an incompressible Newtonian fluid using the no-slip assumption, but have been observed experimentally for blood. Important flow features are also presented that could be used as indicators to observe compressibility and slip in experimental data where near-wall data may be difficult to obtain. Finally, we found that the cumulative averaging method used is ideal for steady particle-based flow methods, as macroscopic no-slip is readily obtained using the MPC bounce-back rule. Generally, a small spurious slip is seen using other averaging methods such as weighted spatial averages or averages over several runs, and the bounce-back rule has to be modified so as to achieve macroscopic no-slip. No modifications of the bounce-back rule were required for our simulations.