  A wealth of astrophysical and cosmological observational evidence shows that
the matter content of the universe is made of about 85$\%$ of non-baryonic dark
matter. Huge experimental efforts have been deployed to look for the direct
detection of dark matter via their scattering on target nucleons, their
production in colliders, and their indirect detection via their annihilation
products. Inelastic scattering of high-energy cosmic rays off dark matter
particles populating the Milky Way halo would produce secondary gamma rays in
the final state from the decay of the neutral pions produced in such
interactions, providing a new avenue to probe dark matter properties. We
compute here the sensitivity for H.E.S.S.-like observatory, a
current-generation ground-based Cherenkov telescopes, to the expected gamma-ray
flux from collisions of Galactic cosmic rays and dark matter in the center of
the Milky Way. We also derive sensitivity prospects for the upcoming Cherenkov
Telescope Array (CTA) and Southern Wide-field Gamma-ray Observatory (SWGO). The
expected sensitivity allows us to probe a poorly-constrained range of dark
matter masses so far, ranging from keV to sub-GeV, and provide complementary
constraints on the dark matter-proton scattering cross section traditionally
probed by deep underground direct dark matter experiments.
