I will discuss the rigorous derivation of hydrodynamic limits of the Boltzmann multi-species equation, for Mach and Knudsen numbers vanishing at the same rate. Solutions to the kinetic system are constructed as fluctuations around local non-equilibrium Maxwellian distributions, whose physical observables solve the limit macroscopic model of interest. A general hypocoercive formalism is developed to establish a uniform (with respect to the small diffusion parameter) Cauchy theory for this perturbative setting. I will apply the method to derive the Maxwell-Stefan cross-diffusion system and present numerical results confirming this convergence. To access the IAC building you need to show an ID-card