  Graphs on lattice points are studied whose edges come from a finite set of
allowed moves of arbitrary length. We show that the diameter of these graphs on
fibers of a fixed integer matrix can be bounded from above by a constant. We
then study the mixing behaviour of heat-bath random walks on these graphs. We
also state explicit conditions on the set of moves so that the heat-bath random
walk, a generalization of the Glauber dynamics, is an expander in fixed
dimension.
