  The zero-temperature Casimir-Lifshitz force between two plates moving
parallel to each other at arbitrary constant speed was found in [New J. Phys.
11, 033035 (2009)]. The solution is here generalized to the case where the
plates are at different temperatures. The Casimir-Lifshitz force is obtained by
calculating the electromagnetic stress tensor, using the method employed by
Antezza et al. [Phys. Rev. A 77, 022901 (2008)] for non-moving plates at
different temperatures. The perpendicular force on the plates has contributions
from the quantum vacuum and from the thermal radiation; both of these
contributions are influenced by the motion. In addition to the perpendicular
force, thermal radiation from the moving plates gives rise to a lateral
component of the Casimir-Lifshitz force, an effect with no quantum-vacuum
contribution. The zero-temperature results are reproduced, in particular the
non-existence of a quantum-vacuum friction between the plates.
