A drift-kinetic Eulerian Vlasov code with fluid equations for the ions is used to study the collision-less diffusion of particles and current across a magnetic field for the case of an electron beam injected near the edge of a two-dimensional magnetized plasma slab. The case of a magnetic field tilted with respect to the beam direction at an angle of θ = 10 deg is considered. Test particles diagnostic techniques are used to study the evolution of the phase space at different locations across the plasma slab. We analyze the anomalous diffusion process triggered by the beam-plasma instability and induced in space across the magnetic field by the Kelvin-Helmholtz instability and the velocity space diffusion induced along the magnetic field due to the kinetic effects of the beam-plasma instability. Ir the present slab geometry it is found that the collision-less diffusion coefficients Dy and Dυ‖ describing respectively the anomalous diffusion in physical spaa and in velocity space, are related by the relation Dy = Dυ‖ tan2 θ/ω2ce. This relation, which links the electror dynamics in the x-y real space and in the y-υ‖ phase space, is verified accurately using the test particles diagnostic techniques. The Vlasov code associated with test particles techniques provides a powerful tool to study particle diffusion in space and in phase space, especially in the low-density regions of the distribution function.