  Integrable models provide an exact description for a wide variety of physical
phenomena. For example nested integrable systems contain different species of
interacting particles with a rich phenomenology in their collective behavior,
which is the origin of the unconventional phenomenon of spin-charge separation.
So far, however, most of the theoretical work in the study of non-equilibrium
dynamics of integrable systems has been focusing on models with an elementary
(i.e., not nested) Bethe ansatz. In this work we explicitly investigate quantum
quenches in nested integrable systems, by generalizing the application of the
Quench Action approach. Specifically, we consider the spin-$1$ Lai-Sutherland
model, described, in the thermodynamic limit, by the theory of two different
species of Bethe-ansatz particles, each one forming an infinite number of bound
states. We focus on the situation where the quench dynamics starts from a
simple matrix product state for which the overlaps with the eigenstates of the
Hamiltonian are known. We fully characterize the post-quench steady state and
perform several consistency checks for the validity of our results. Finally, we
provide predictions for the propagation of entanglement and mutual information
after the quench, which can be used as signature of the quasi-particle content
of the model.
