  In this third paper of a series, we study the growth and luminosity of black
holes (BHs) in motion with respect to their surrounding medium. We run a large
set of two-dimensional (2D) axis-symmetric simulations to explore a large
parameter space of initial conditions and formulate an analytical model for the
accretion. Contrary to the case without radiation feedback, the accretion rate
increases with increasing BH velocity $\vbh$ reaching a maximum value at
$\vbh=2c_{\rm s,in}\sim 50$ km/s, where $c_{\rm s,in}$ is the sound speed
inside the "cometary-shaped" \hii region around the BH, before decreasing as
$\vbh^{-3}$ when the I-front becomes {\it R}-type (rarefied) and the accretion
rate approaches the classical Bondi-Hoyle-Lyttleton solution. The increase of
the accretion rate with $\vbh$ is produced by the formation of a {\it D}-type
(dense) ionization front (I-front) preceded by a standing bow-shock that
reduces the downstream gas velocity to transonic values. There is a range of
densities and velocities where the dense shell is unstable producing periodic
accretion rate peaks which can significantly increase the detectability of
intermediate mass BHs. We find that the mean accretion rate for a moving BH is
larger than that of a stationary BH of the same mass if the medium temperature
is $T_\infty<10^4$ K. This result could be important for the growth of seed BHs
in the multi-phase medium of the first galaxies and for building and early
X-ray background that may affect the formation of the first galaxies and the
reionization process.
