For rank-two bundles on genus-two hyperelliptic curves, the Chow ring of the universal moduli stack is tautological with explicit generators and relations.
Heinloth, Lectures on the M oduli S tack of V ector B undles on a C urve , Affine Flag Manifolds and Principal Bundles (A.Schmitt, ed.)
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Cycles in the universal moduli stack of bundles of rank two over genus two curves
For rank-two bundles on genus-two hyperelliptic curves, the Chow ring of the universal moduli stack is tautological with explicit generators and relations.