  We consider quantum or classical many-body Hamiltonian systems, whose
dynamics is given by an integrable, contact interactions, plus another,
possibly long-range, generic two-body potential. We show how the dynamics of
local observables is given in terms of a generalised version of
Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, which we denote as gBBGKY, which
is built for the densities, and their correlations, of the quasiparticles of
the underlying integrable model. Unlike the usual cases of perturbation theory
from free gases, the presence of local interactions in the integrable model
"lifts" the so-called kinetic blocking, and the second layer of the hierarchy
reproduces the dynamics at all time-scales. The latter consists of a fast
pre-equilibration to a non-thermal steady state, and its subsequent
thermalisation to a Gibbs ensemble. We show how the final relaxation is encoded
into a Boltzmann scattering integral involving three or higher
body-scatterings, and which, remarkably, is entirely determined by the
diffusion constants of the underlying integrable model. We check our results
with exact molecular dynamics simulations, finding perfect agreement. Our
results show how gBBGKY can be successfully employed in quantum systems to
compute scattering integrals and Fermi's golden rule transition rates.
