  The concept of the generalized continuity equation (GCE) was recently
introduced in [J. Phys. A: Math. and Theor. {\bf 52}, 1552034 (2019)], and was
derived in the context of $N$ independent Schr\"{o}dinger systems. The GCE is
induced by a symmetry transformation which mixes the states of these systems,
even though the $N$-system Lagrangian does not. As the $N$-system
Schr\"{o}dinger Lagrangian is not invariant under such a transformation, the
GCE will involve source terms which, under certain conditions vanish and lead
to conserved currents. These conditions may hold globally or locally in a
finite domain, leading to globally or locally conserved currents, respectively.
In this work, we extend this idea to the case of arbitrary
$SU(N)$-transformations and we show that a similar GCE emerges for $N$ systems
in the Dirac dynamics framework. The emerging GCEs and the conditions which
lead to the attendant conservation laws provide a rich phenomenology and
potential use for the preparation and control of fermionic states.
