  It has been long-understood that final state rescattering effects provide
$\mathcal{O}(1)$ corrections to hadronic meson decays rates, such as
$\eta\to\pi\pi\pi$ and $\eta^{\prime}\to\eta\pi\pi$. Hence, one would expect
that such effects would be just as important in axio-hadronic $\eta$ and
$\eta^{\prime}$ decays, such as $\eta^{(\prime)}\to\pi\pi a$, where $a$ is an
axion or axion-like particle (ALP). And indeed they are, as we show in this
paper by using the treatment of dispersion relations to include the effects of
strong final state interactions in several axio-hadronic processes, namely,
$\eta^{(\prime)}\to\pi^0\pi^0 a$, $\eta^{(\prime)}\to\pi^+\pi^- a$, and
$\eta^{\prime}\to\eta\pi^0 a$. We also compute the perturbative, leading order
decay rates for multiple ALP emission, such as in
$\eta^{(\prime)}\to\pi^{0}aa$, $\eta^{\prime}\to\eta aa$ and
$\eta^{(\prime)}\to aaa$, and briefly discuss the expected corrections from
strong interactions and the processes that must be considered for an accurate
rate estimation of these multi-ALP decay channels.
