  This paper presents a systematic and accurate treatment of the evolution of
cosmological perturbations in self-interacting dark matter models, for
particles which decoupled from the primordial plasma while relativistic. We
provide a numerical implementation of the Boltzmann hierarchies developed in a
previous paper [JCAP, 09 (2020) 041] in a publicly available Boltzmann code and
show how it can be applied to realistic DM candidates such as sterile neutrinos
either under resonant or non-resonant production mechanisms, and for different
field mediators. At difference with traditional fluid approximations -- also
known as a $c_{\rm eff}-c_{\rm vis}$ parametrizations -- our approach follows
the evolution of phase-space perturbations under elastic DM interactions for a
wide range of interaction models, including the effects of late kinetic
decoupling. Finally, we analyze the imprints left by different self interacting
models on linear structure formation, which can be constrained using
Lyman-$\alpha$ forest and satellite counts. We find new lower bounds on the
particle mass that are less restrictive than previous constraints.
