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

Ramsey expansions of $\Lambda$-ultrametric spaces