  The interaction between drifting carriers and traveling electromagnetic waves
is considered within the context of the classical Boltzmann transport equation
to compute the Casimir-Lifshitz force between media with small density of
charge carriers, including dielectrics and intrinsic semiconductors. We expand
upon our previous work [Phys. Rev. Lett. {\bf 101}, 163203 (2008)] and derive
in some detail the frequency-dependent reflection amplitudes in this theory and
compute the corresponding Casimir free energy for a parallel plate
configuration. We critically discuss the the issue of verification of the
Nernst theorem of thermodynamics in Casimir physics, and explicity show that
our theory satisfies that theorem. Finally, we show how the theory of drifting
carriers connects to previous computations of Casimir forces using spatial
dispersion for the material boundaries.
