We discuss how LHC di-muon data collected to study B-q -> mu mu can be used to constrain light particles with flavour-violating couplings to b-quarks. Focussing on the case of a flavoured QCD axion, a, we compute the decay rates for B-q -> mu mu a and the SM background process B-q -> mu mu gamma near the kinematic endpoint. These rates depend on non-perturbative B-q -> gamma(()*()) form factors with on- or off-shell photons. The off-shell form factors - relevant for generic searches for beyond-the-SM particles - are discussed in full generality and computed with QCD sum rules for the first time. This includes an extension to the low-lying resonance region using a multiple subtracted dispersion relation. With these results, we analyse available LHCb data to obtain the sensitivity on B-q -> mu mu a at present and future runs. We find that the full LHCb dataset alone will allow to probe axion-coupling scales of the order of 10(6) GeV for both b -> d and b -> s transitions. As a spin-off application of the off-shell form factors we further analyse the case of light, Beyond the Standard Model, vectors.