We study slope-stable vector bundles and Bridgeland stability conditions on varieties which are a quotient of a smooth projective variety by a finite abelian group $G$ acting freely. We show there is an analytic isomorphism between $G$-invariant geometric stability conditions on the cover and geometric stability conditions on the quotient that are invariant under the residual action of the group $\widehat{G}$ of irreducible representations of $G$. We apply our results to describe a connected component inside the stability manifolds of free abelian quotients when the cover has finite Albanese morphism. This applies to varieties with non-finite Albanese morphism which are free abelian quotients of varieties with finite Albanese morphism, such as Beauville-type and bielliptic surfaces. This gives a partial answer to a question raised by Lie Fu, Chunyi Li, and Xiaolei Zhao: if a variety $X$ has non-finite Albanese morphism, does there always exist a non-geometric stability condition on $X$? We also give counterexamples to a conjecture of Fu--Li--Zhao concerning the Le Potier function, which characterises Chern classes of slope-semistable sheaves. As a result of independent interest, we give a description of the set of geometric stability conditions on an arbitrary surface in terms of a refinement of the Le Potier function. This generalises a result of Fu--Li--Zhao from Picard rank $1$ to arbitrary Picard rank.

39 pages, final version to appear in Épijournal de Géométrie Algébrique