In recent works it has been demonstrated that using an appropriate rescaling, linear kinetic equations with heavy tailed equilibria give rise to a scalar fractional diffusion equation. In this talk an extension of this is presented, where the linear kinetic equations under consideration, not only conserves mass, but also momentum and energy. In the limit, fractional diffusion equations are obtained for the energy and the mass, while the equation for the momentum is trivial. The methods of proof presented rely on spectral analysis combined with energy estimates. It is constructive and provides explicit convergence rates. This is work in progress together with É. Bouin and C. Mouhot.