  Dark matter interacting via the exchange of a light pseudoscalar can induce
observable signals in indirect detection experiments and experience large
self-interactions while evading the strong bounds from direct dark matter
searches. The pseudoscalar mediator will however induce flavour-changing
interactions in the Standard Model, providing a promising alternative way to
test these models. We investigate in detail the constraints arising from rare
meson decays and fixed target experiments for different coupling structures
between the pseudoscalar and Standard Model fermions. The resulting bounds are
highly complementary to the information inferred from the dark matter relic
density and the constraints from primordial nucleosynthesis. We discuss the
implications of our findings for the dark matter self-interaction cross section
and the prospects of probing dark matter coupled to a light pseudoscalar with
direct or indirect detection experiments. In particular, we find that a
pseudoscalar mediator can only explain the Galactic Centre excess if its mass
is above that of the B mesons, and that it is impossible to obtain a
sufficiently large direct detection cross section to account for the DAMA
modulation
