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We prove formulas for the rational Chow motives of moduli spaces ofsemistable vector bundles and Higgs bundles of rank 3 and coprime degree on asmooth projective curve. Our approach involves identifying criteria to liftidentities in (a completion of) the Grothendieck group of effective Chowmotives to isomorphisms in the category of Chow motives. For the Higgs modulispace, we use motivic Bialynicki-Birula decompositions associated to a scalingaction with variation of stability and wall-crossing for moduli spaces of rank2 pairs, which occur in the fixed locus of this action.

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