For rank-two bundles on genus-two hyperelliptic curves, the Chow ring of the universal moduli stack is tautological with explicit generators and relations.
Zagier, On the C ohomology of M oduli S paces of R ank T wo V ector B undles O ver C urves , The Moduli Space of Curves
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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.