  The past few years have witnessed the development of a comprehensive theory
to describe integrable systems out of equilibrium, in which the Bethe ansatz
formalism has been tailored to address specific problems arising in this
context. While most of the work initially focused on the study of prototypical
models such as the well-known Heisenberg chain, many theoretical results have
been recently extended to a class of more complicated $\it{nested}$ integrable
systems, displaying different species of quasiparticles. Still, in the simplest
context of quantum quenches, the vast majority of theoretical predictions have
been numerically verified only in systems with an elementary Bethe ansatz
description. In this work, we fill this gap and present a direct numerical test
of some results presented in the recent literature for nested systems, focusing
in particular on the Lai-Sutherland model. Using time-dependent density matrix
renormalization group and exact diagonalization methods, we compute the
spreading of both correlation functions and entanglement entropy after a quench
from a simple class of product initial states. This allows us to test the
validity of the nested version of a conjectured formula, based on the
quasiparticle picture, for the growth of the entanglement entropy, and the
Bethe ansatz predictions for the "light-cone" velocity of correlation
functions.
