Axion models with generation-dependent Peccei-Quinn charges can lead to flavor-changing neutral currents, thus motivating QCD axion searches at precision flavor experiments. We rigorously derive limits on the most general effective flavor-violating couplings from current measurements and assess their discovery potential. For two-body decays, we use available experimental data to derive limits on q →q'a decay rates for all flavor transitions. Axion contributions to neutral-meson mixing are calculated in a systematic way using chiral perturbation theory and operator product expansion. We also discuss in detail baryonic decays and three-body meson decays, which can lead to the best search strategies for some of the couplings. For instance, a strong limit on the Λ →n a transition can be derived from the supernova SN 1987A. In the near future, dedicated searches for q →q'a decays at ongoing experiments could potentially test Peccei-Quinn breaking scales up to 1012 GeV at NA62 or KOTO and up to 109 GeV at Belle II or BES III.