  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 Bialynicki-Birula decompositions associated to a scaling
action with variation of stability and wall-crossing for moduli spaces of rank
2 pairs, which occur in the fixed locus of this action.
