Motives of moduli spaces of rank $3$ vector bundles and Higgs bundles on a curve

We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank $3$ and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Białynicki-Birula decompositions associated with a scaling action, together with the variation of stability and wall-crossing for moduli spaces of rank $2$ pairs, which occur in the fixed locus of this action.