  We afford a systematic and comprehensive account of the canonical dynamics of
4D Regge Calculus perturbatively expanded to linear order around a flat
background. To this end, we consider the Pachner moves which generate the most
basic and general simplicial evolution scheme. The linearized regime features a
vertex displacement (`diffeomorphism') symmetry for which we derive an abelian
constraint algebra. This permits to identify gauge invariant `lattice
gravitons' as propagating curvature degrees of freedom. The Pachner moves admit
a simple method to explicitly count the gauge and `graviton' degrees of freedom
on an evolving triangulated hypersurface and we clarify the distinct role of
each move in the dynamics. It is shown that the 1-4 move generates four `lapse
and shift' variables and four conjugate vertex displacement generators; the 2-3
move generates a `graviton'; the 3-2 move removes one `graviton' and produces
the only non-trivial equation of motion; and the 4-1 move removes four `lapse
and shift' variables and trivializes the four conjugate symmetry generators. It
is further shown that the Pachner moves preserve the vertex displacement
generators. These results may provide new impetus for exploring `graviton
dynamics' in discrete quantum gravity models.
