  We find a near detailed balance solution to the relativistic Boltzmann
equation under the relaxation time approximation with a collision term which
differs from the Anderson-Witting model and is dependent on the stationary
observer. Using this new solution, we construct an explicit covariant transport
equation for the particle flux in response to the generalized temperature and
chemical potential gradients in generic stationary spacetimes, with the
transport tensors characterized by some integral functions in the chemical
potential and the relativistic coldness. To illustrate the application of the
transport equation we study probe systems in Rindler and Kerr spacetimes and
analyze the asymptotic properties of the gravito-conductivity tensor in the
near horizon limit. It turns out that both the longitudinal and lateral parts
(if present) of the gravito-conductivity tend to be divergent in the near
horizon limit. In the weak field limit, our results coincide with with the
non-relativistic gravitational transport equation which follows from the direct
application of the Drude model.
