Solar wind plasma temperature and solar wind turbulence at ion and electron scales often show anisotropic features, with different temperature and fluctuation power in parallel and perpendicular direction with respect to the orientation of the background magnetic field. The ratio between the power of the magnetic field fluctuations in parallel and perpendicular direction at the ion scales may vary with the heliospheric distance and depends on various parameters, such as the plasma compressibility, waves properties and the non-thermal plasma features, such as temperature anisotropies and relative drift speeds. In this work we have performed 2.5D hybrid simulations to study the importance of relative drifts and a gradual solar wind expansion in a multi-species plasma, consisting of fluid electrons, kinetic (particle-in-cell) protons and a drifting population of He ++ ions. At the beginning of the simulations we impose a turbulent spectrum of paralllel propagating Alfv\'en-cyclotron waves, co-existing with the drifting multi-species plasma. In the course of nonlinear evolution of the system we observe substaintial anisotropic cascade of the magnetic field power spectra towards perpendicular wave numbers. The nature of the anisotropic turbulent cascade depends on the differential streaming between the different ion populations and is affected by the solar wind expansion. In the case of sub-Alfv\'enic differential streaming the perpendicular wave power is enhanced and the anisotropic cascade is shifted towards smaller wave numbers.