  This contribution deals with $\mathrm L^2$ hypocoercivity methods for kinetic
Fokker-Planck equations with integrable local equilibria and a
\emph{factorisation} property that relates the Fokker-Planck and the transport
operators. Rates of convergence in presence of a global equilibrium, or decay
rates otherwise, are estimated either by the corresponding rates in the
diffusion limit, or by the rates of convergence to local equilibria, under
moment conditions. On the basis of the underlying functional inequalities, we
establish a classification of decay and convergence rates for large times,
which includes for instance sub-exponential local equilibria and
sub-exponential potentials.
