  We prove that in every compact space of Delone sets in $\mathbb{R}^d$ which
is minimal with respect to the action by translations, either all Delone sets
are uniformly spread, or continuously many distinct bounded displacement
equivalence classes are represented, none of which contains a lattice. The
implied limits are taken with respect to the Chabauty--Fell topology, which is
the natural topology on the space of closed subsets of $\mathbb{R}^d$. This
topology coincides with the standard local topology in the finite local
complexity setting, and it follows that the dichotomy holds for all minimal
spaces of Delone sets associated with well-studied constructions such as
cut-and-project sets and substitution tilings, whether or not finite local
complexity is assumed.
