  The "dark photon" $\gamma_d$ of a gauged $U(1)_d$ can become practically
invisible if it primarily decays into light states from a dark sector. We point
out that, in such scenarios, the running of the $U(1)_d$ "fine structure
constant" $\alpha_d$, with momentum transfer $q^2$, can be significant and
potentially measurable. The $\gamma_d$ kinetic mixing parameter $\varepsilon^2$
is also expected to run with $q^2$, through its dependence on $\alpha_d$. We
show how the combined running of $\varepsilon^2 \alpha_d$ may provide a probe
of the spectrum of dark particles and, for $\alpha_d\gtrsim {\rm few}\times
0.1$, substantially modify predictions for "beam dump" or other intense source
experiments. These features are demonstrated in simple models that contain
light dark matter and a scalar that breaks $U(1)_d$. We also discuss theoretic
considerations, regarding the $U(1)_d$ model in the ultraviolet regime, that
may suggest the infrared upper bound $\alpha_d \lesssim 0.1$.
