  We prove a version of the moving particle lemma for the exclusion process on
any finite weighted graph, based on the octopus inequality of Caputo, Liggett,
and Richthammer. In light of their proof of Aldous' spectral gap conjecture, we
conjecture that our moving particle lemma is optimal in general. Our result can
be applied to graphs which lack translational invariance, including, but not
limited to, fractal graphs. An application of our result is the proof of local
ergodicity for the exclusion process on a class of weighted graphs, the details
of which are reported in a follow-up paper [arXiv:1705.10290].
