The Galvin-Prikry theorem, an infinite-dimensional generalization of the infinite Ramsey theorem, can be stated in the language of semigroups as follows: for any finite Borel partition of the semigroup of monotone injections from $\omega$ into itself, one piece contains a right ideal. This talk provides a generalization of this result to the semigroups of embeddings of certain more complicated first-order structures. The definition and construction of these structures arise from the study of big Ramsey degrees in binary free amalgamation classes. This is joint work with Natasha Dobrinen.