  We derive an analytical approximation for the linear scaling evolution of the
characteristic length $L$ and the root-mean-squared velocity $\sigma_v$ of
standard frictionless domain wall networks in
Friedmann-Lema\^itre-Robertson-Walker universes with a power law evolution of
the scale factor $a$ with the cosmic time $t$ ($a \propto t^\lambda$). This
approximation, obtained using a recently proposed parameter-free
velocity-dependent one-scale model for domain walls, reproduces well the model
predictions for $\lambda$ close to unity, becoming exact in the $\lambda \to
1^-$ limit. We use this approximation, in combination with the exact results
found for $\lambda=0$, to obtain a fit to the model predictions valid for
$\lambda \in [0, 1[$ with a maximum error of the order of $1 \%$. This fit is
also in good agreement with the results of field theory numerical simulations,
specially for $\lambda \in [0.9, 1[$. Finally, we explicitly show that the
phenomenological energy-loss parameter of the original velocity-dependent
one-scale model for domain walls vanishes in the $\lambda \to 1^-$ limit and
discuss the implications of this result.
