For rank-two bundles on genus-two hyperelliptic curves, the Chow ring of the universal moduli stack is tautological with explicit generators and relations.
Fringuelli, The P icard group of the universal moduli space of vector bundles on stable curves , Advances in Mathematics 336 (2018), 477–557
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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.