  We propose a unified method for the large space-time scaling limit of
\emph{linear} collisional kinetic equations in the whole space. The limit is of
\emph{fractional} diffusion type for heavy tail equilibria with slow enough
decay, and of diffusive type otherwise. The proof is constructive and the
fractional/standard diffusion matrix is obtained. The method combines energy
estimates and quantitative spectral methods to construct a `fluid mode'. The
method is applied to scattering models (without assuming detailed balance
conditions), Fokker-Planck operators and L{\'e}vy-Fokker-Planck operators. It
proves a series of new results, including the fractional diffusive limit for
Fokker-Planck operators in any dimension, for which the formulas for the
diffusion coefficient were not known, for L{\'e}vy-Fokker-Planck operators with
general equilibria, and for scattering operators including some cases of
infinite mass equilibria. It also unifies and generalises the results of
previous papers with a quantitative method, and our estimates on the fluid
approximation error also seem novel.
