  Several quantum gravity approaches and field theory on an evolving lattice
involve a discretization changing dynamics generated by evolution moves. Local
evolution moves in variational discrete systems (1) are a generalization of the
Pachner evolution moves of simplicial gravity models, (2) update only a small
subset of the dynamical data, (3) change the number of kinematical and physical
degrees of freedom, and (4) generate a dynamical coarse graining or refining of
the underlying discretization. To systematically explore such local moves and
their implications in the quantum theory, this article suitably expands the
quantum formalism for global evolution moves, constructed in a companion paper,
by employing that global moves can be decomposed into sequences of local moves.
This formalism is spelled out for systems with Euclidean configuration spaces.
Various types of local moves, the different kinds of constraints generated by
them, the constraint preservation and possible divergences in resulting state
sums are discussed. It is shown that non-trivial local coarse graining moves
entail a non-unitary projection of (physical) Hilbert spaces and `fine grained'
Dirac observables defined on them. Identities for undoing a local evolution
move with its (time reversed) inverse are derived. Finally, the implications of
these results for a Pachner move generated dynamics in simplicial quantum
gravity models are commented on.
