  We perform a detailed comparison between a recently proposed parameter-free
velocity-dependent one-scale model and the standard parametric model for the
cosmological evolution of domain wall networks. We find that the latter
overestimates the damping of the wall motion due to the Hubble expansion and
neglects the direct impact of wall decay on the evolution of the
root-mean-square velocity of the network. We show that these effects are
significant but may be absorbed into a redefinition of the momentum parameter.
We also discuss the implications of these findings for cosmic strings. We
compute the energy loss and momentum parameters of the standard parametric
model for cosmological domain wall evolution using our non-parametric
velocity-dependent one-scale model in the context of cosmological models having
a power law evolution of the scale factor $a$ with the cosmic time $t$ ($a
\propto t^\lambda$, $0 < \lambda < 1$), and compare with the results obtained
from numerical field theory simulations. We further provide simple linear
functions which roughly approximate the dependence of the energy loss and
momentum parameters on $\lambda$.
