In the harmonic gauge, the role of the constraint equations as
such is hidden. This is because the constraint equations become
evolution equations due to the gauge choice. Indeed, in the full
harmonic gauge all components of the Einstein tensor become a big
second order hyperbolic equation for the metric components. The
need for the constraints to be satisfied at the initial surface
enters the picture because it implies that the first time
derivative of the gauge condition must vanish there in order to
guarantee that the constraints vanish everywhere. Thus the
question about the preservation of the constraint equations does
not appear here. At most one can say it has been traded for the
question of the gauge consistency.