  We study single-flip dynamics in sets of three-dimensional rhombus tilings
with fixed polyhedral boundaries. This dynamics is likely to be slowed down by
so-called ``cycles'': such structures arise when tilings are encoded via the
``partition-on-tiling'' method and are susceptible to break connectivity by
flips or at least ergodicity, because they locally suppress a significant
amount of flip degrees of freedom. We first address the so-far open question of
the connectivity of tiling sets by elementary flips. We prove exactly that sets
of tilings of codimension one and two are connected for any dimension and
tiling size. For higher-codimension tilings of dimension 3, the answer depends
on the precise choice of the edge orientations, which is a non-trivial issue.
In most cases, we can prove connectivity despite the existence of cycles. In
the few remaining cases, among which the icosahedral symmetry, the question
remains open. We also study numerically flip-assisted diffusion to explore the
possible effects of the previously mentioned cycles. Cycles do not seem to slow
down significantly the dynamics, at least as far as self-diffusion is
concerned.
