  The quantum kinetic equation for the gauge-invariant Wigner function,
constructed from spinor fields that obey the Dirac equation modified by CPT and
Lorentz symmetry-violating terms, is presented. The equations for the
components of Wigner function in the Clifford algebra basis are accomplished.
Focusing on the massless case, an extended semiclassical chiral kinetic theory
in the presence of external electromagnetic fields is developed. We calculate
the chiral currents and establish the anomalous magnetic and separation effects
in a Lorentz-violating background. The chiral anomaly within the context of
extended Quantum Electrodynamics is elucidated. Finally, we derive the
semiclassical Lorentz-violating extended chiral transport equation.
