  We analyze a new class of FCNC processes, the $f \to f^{\prime} \,
\bar{\gamma}$ decays of a fermion $f$ into a lighter (same-charge) fermion
$f^{\prime}$ plus a {\it massless} neutral vector boson, a {\it dark photon}
$\bar{\gamma}$. A massless dark photon does not interact at tree level with
observable fields, and the $f \!\to\! f^{\prime} \, \bar{\gamma}$ decay
presents a characteristic signature where the final fermion $f^{\prime}$ is
balanced by a {\it massless invisible} system. Models recently proposed to
explain the exponential spread in the standard-model Yukawa couplings can
indeed foresee an extra unbroken {\it dark} $U(1)$ gauge group, and the
possibility to couple on-shell dark photons to standard-model fermions via
one-loop magnetic-dipole kind of FCNC interactions. The latter are suppressed
by the characteristic scale related to the mass of heavy messengers, connecting
the standard model particles to the dark sector. We compute the corresponding
decay rates for the top, bottom, and charm decays ($t\to c\, \bar{\gamma},u\,
\bar{\gamma}$, $\;b\to s\, \bar{\gamma},d\, \bar{\gamma}$, and $c\to u
\bar{\gamma}$), and for the charged-lepton decays ($\tau \to \mu\,
\bar{\gamma}, e\, \bar{\gamma}$, and $\mu \to e \bar{\gamma}$) in terms of
model parameters. We find that large branching ratios for both quark and lepton
decays are allowed in case the messenger masses are in the discovery range of
the LHC. Implications of these new decay channels at present and future
collider experiments are briefly discussed.
