We consider the flow of blood, treated as an incompressible Newtonian fluid, through vessels undergoing periodic oscillations. As remarked by many authors, in the absence of valves oscillations hinder the flow because of the lumen reduction. The underlying biological mechanism is the so-called vasomotion, observed long ago in small blood vessels. Here, we study the vasomotion in arterioles and provide its theoretical justification by analyzing the effect when the network of vessels downstream of the arterioles is considered. We thus explain both quantitatively and qualitatively, why the oscillations of the arteriole walls, a phenomenon that undoubtedly reduces blood flow at the level of the single arteriole, play a fundamental role in microcirculation. In "large" arterioles we analyze also the coupling between the vasomotion and the Fåhræus-Lindqvist effect (the tendency of the erythrocytes to accumulate towards the center). In particular, we prove that the presence of a cell depleted layer close to the vessel walls mitigates the disadvantage caused by the lumen reduction.