  For a finite lattice $\Lambda$, $\Lambda$-ultrametric spaces are a convenient
language for describing structures equipped with a family of equivalence
relations. When $\Lambda$ is finite and distributive, there exists a generic
$\Lambda$-ultrametric space, and we here identify a family of Ramsey expansions
for that space. This then allows a description the universal minimal flow of
its automorphism group, and also implies the Ramsey property for all known
homogeneous finite-dimensional permutation structures, i.e. structures in a
language of finitely many linear orders. A point of technical interest is that
our proof involves classes with non-unary algebraic closure operations. As a
byproduct of some of the concepts developed, we also arrive at a natural
description of the known homogeneous finite-dimensional permutation structures,
completing our previously begun "census".
