Context: Galactic black hole coronae are composed of a hot, magnetized plasma. The spectral energy distribution produced in this component of X-ray binaries can be strongly affected by different interactions between locally injected relativistic particles and the matter, radiation and magnetic fields in the source. Aims: We study the non-thermal processes driven by the injection of relativistic particles into a strongly magnetized corona around an accreting black hole. Methods: We compute in a self-consistent way the effects of relativistic bremsstrahlung, inverse Compton scattering, synchrotron radiation, and the pair-production/annihilation of leptons, as well as hadronic interactions. Our goal is to determine the non-thermal broadband radiative output of the corona. The set of coupled kinetic equations for electrons, positrons, protons, and photons are solved and the resulting particle distributions are computed self-consistently. The spectral energy distributions of transient events in X-ray binaries are calculated, as well as the neutrino production. Results: We show that the application to Cygnus X-1 of our model of non-thermal emission from a magnetized corona yields a good fit to the observational data. Finally, we show that the accumulated signal produced by neutrino bursts in black hole coronae might be detectable for sources within a few kpc on timescales of years. Conclusions: Our work leads to predictions for non-thermal photon and neutrino emission generated around accreting black holes, that can be tested by the new generation of very high energy gamma-ray and neutrino instruments.