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 equilibria satisfy a {em generalised} weighted mass condition and can have infinite mass. 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 characterization of the diffusion coefficient was not known, for L{e}vy-Fokker-Planck operators with general equilibria, and in cases where the equilibrium has infinite mass. It also unifies and generalises the results of ten previous papers with a quantitative method, and our estimates on the fluid approximation error seem novel in these cases.
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