  Ho\v{r}ava gravity breaks Lorentz symmetry by introducing a preferred
spacetime foliation, which is defined by a timelike dynamical scalar field, the
khronon. The presence of this preferred foliation makes black hole solutions
more complicated than in General Relativity, with the appearance of multiple
distinct event horizons: a matter horizon for light/matter fields, a spin-0
horizon for the scalar excitations of the khronon, a spin-2 horizon for
tensorial gravitational waves, and even, at least in spherical symmetry, a
universal horizon for instantaneously propagating modes appearing in the
ultraviolet. We study how black hole solutions in Ho\v{r}ava gravity change
when the black hole is allowed to move with low velocity relative to the
preferred foliation. These slowly moving solutions are a crucial ingredient to
compute black hole "sensitivities" and predict gravitational wave emission (and
in particular dipolar radiation) from the inspiral of binary black hole
systems. We find that for generic values of the theory's three dimensionless
coupling constants, slowly moving black holes present curvature singularities
at the universal horizon. Singularities at the spin-0 horizon also arise unless
one waives the requirement of asymptotic flatness at spatial infinity.
Nevertheless, we have verified that at least in a one-dimensional subset of the
(three-dimensional) parameter space of the theory's coupling constants, slowly
moving black holes are regular everywhere, even though they coincide with the
general relativistic ones (thus implying in particular the absence of dipolar
gravitational radiation). Remarkably, this subset of the parameter space
essentially coincides with the one selected by the recent constraints from
GW170817 and by solar system tests.
