  The motion of electron wave packets in the Dirac semimetals A$_3$Bi
(A=Na,K,Rb) is studied in a semiclassical approximation. Because of the
two-fold degeneracy of the Dirac points and a momentum-dependent gap term in
the low-energy Hamiltonian, the associated Berry curvature is non-Abelian. In
the presence of background electromagnetic fields, such a Berry curvature leads
to a splitting of trajectories for the wave packets that originate from
different Dirac points and chiral sectors. The nature of the splitting strongly
depends on the background fields as well as the initial chiral composition of
the wave packets. In parallel electric and magnetic fields, while a well
pronounced valley splitting is achieved for any chirality composition, the
chirality separation takes place predominantly for the initially polarized
states. On the other hand, in perpendicular electric and magnetic fields, there
are clear deviations from the conventional Abelian trajectories, albeit without
a well-pronounced valley splitting.
