  We present a real-time lattice approach to study the non-equilibrium dynamics
of vector and axial charges in $SU(N) \times U(1)$ gauge theories. Based on a
classical description of the non-Abelian and Abelian gauge fields, we include
dynamical fermions and develop operator definitions for (improved) Wilson and
overlap fermions that allow us to study real-time manifestations of the axial
anomaly from first principles. We present a first application of this approach
to anomalous transport phenomena such as the Chiral Magnetic Effect (CME) and
Chiral Separation Effect (CSE) by studying the dynamics of fermions during and
after a $SU(N)$ sphaleron transition in the presence of a $U(1)$ magnetic
field. We investigate the fermion mass and magnetic field dependence of the
suggested signatures of the CME and CSE and point out some important aspects
which need to be accounted for in the macroscopic description of anomalous
transport phenomena.
