  We continue our exploration of the wiggly generalisation of the
Velocity-Dependent One Scale Model for cosmic strings, through the study of its
allowed asymptotic scaling solutions. We extend the work of a previous paper
[Almeida $\&$ Martins, Phys. Rev. D 104 (2021) 043524] by considering the more
comprehensive case of a time-varying coarse-graining scale for the string
wiggles. The modeling of the evolution of the network therefore relies on three
main mechanisms: Hubble expansion, energy transfer mechanisms (e.g., the
production of loops and wiggles) and the choice of the scale at which wiggles
are coarse-grained. We analyse the role of each of them on the overall
behaviour of the network, and thus in the allowed scaling solutions. In
Minkowski space, we find that linear scaling, previously observed in numerical
simulations without expansion, is not possible with a changing averaging scale.
For expanding universes, we find that the three broad classes of scaling
solutions -- with the wiggliness disappearing, reaching scaling, or growing --
still exist but are differently impacted by the time evolution of the
coarse-graining scale. Nambu-Goto type solutions (without wiggles) are
unaffected, growing wiggliness solutions are trivially generalized, while for
solutions where wiggliness reaches scaling the expansion rate for which the
solution exists is decreased with respect to the one for a fixed
coarse-graining scale. Finally, we also show that the inclusion of a
time-varying coarse-graining scale allows, in principle, for additional scaling
solutions which, although mathematically valid, are not physical. Overall, our
mapping of the landscape of the allowed scaling solutions of the wiggly
Velocity-Dependent One Scale Model paves the way for the detailed testing of
the model, to be done by forthcoming high-resolution field theory and
Nambu-Goto simulations.
