  We compute the Coulomb branch partition function of the 4d $\mathcal{N}=2$
vector multiplet on closed simply-connected quasi-toric manifolds $B$. This
includes a large class of theories, localising to either instantons or
anti-instantons at the torus fixed points (including Donaldson-Witten and
Pestun-like theories as examples). The main difficulty is to obtain flux
contributions from the localisation procedure. We achieve this by taking a
detour via the 5d $\mathcal{N}=1$ vector multiplet on closed simply-connected
toric Sasaki-manifolds $M$ which are principal $S^1$-bundles over $B$. The
perturbative partition function can be expressed as a product over slices of
the toric cone. By taking finite quotients $M/\mathbb{Z}_h$ along the $S^1$,
the locus picks up non-trivial flat connections which, in the limit
$h\to\infty$, provide the sought-after fluxes on $B$. We compute the one-loop
partition functions around each topological sector on $M/\mathbb{Z}_h$ and $B$
explicitly, and then factorise them into contributions from the torus fixed
points. This enables us to also write down the conjectured instanton part of
the partition function on $B$.
