  We present a semiclassical analysis for Dirac fields on an arbitrary
spacetime background and in the presence of a fixed electromagnetic field. Our
approach is based on a Wentzel-Kramers-Brillouin approximation, and the results
are analyzed at leading and next-to-leading order in the small expansion
parameter $\hbar$. Taking into account the spin-orbit coupling between the
internal and external degrees of freedom of wave packets, we derive effective
ray equations with spin-dependent terms. These equations describe the
gravitational spin Hall effect of localized Dirac wave packets. We treat both
massive and massless Dirac fields and show how a covariantly defined Berry
connection and the associated Berry curvature govern the semiclassical
dynamics. The gravitational spin Hall equations are shown to be particular
cases of the Mathisson-Papapetrou equations for spinning objects.
