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

Scaling solutions of wiggly cosmic strings: II. Time-varying coarse-graining scale solutions