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 $q\to q' a$ 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 $��\to n a$ transition can be derived from the supernova SN 1987A. In the near future, dedicated searches for $q\to q' a$ decays at ongoing experiments could potentially test Peccei-Quinn breaking scales up to $10^{12}$ GeV at NA62 or KOTO, and up to $10^{9}$ GeV at Belle II or BES III.

23 pages, 4 tables, 4 figures