  This work considers entropy generation and relaxation in quantum quenches in
the Ising and $3$-state Potts spin chains. In the absence of explicit symmetry
breaking we find universal ratios involving R\'enyi entropy growth rates and
magnetisation relaxation for small quenches. We also demonstrate that the
magnetisation relaxation rate provides an observable signature for the
"dynamical Gibbs effect" which is a recently discovered characteristic
non-monotonous behaviour of entropy growth linked to changes in the
quasi-particle spectrum.
