Every Borel-overt fiberwise quasi-Polish groupoid is Borel equivalent to an open Polish groupoid, hence to an action groupoid of a Polish group action; Borel-overt classwise Polish equivalence relations are Borel bireducible with orbit equivalence relations of free Polish group actions.
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Componentwise Polish groupoids and equivalence relations
Every Borel-overt fiberwise quasi-Polish groupoid is Borel equivalent to an open Polish groupoid, hence to an action groupoid of a Polish group action; Borel-overt classwise Polish equivalence relations are Borel bireducible with orbit equivalence relations of free Polish group actions.