More on the Kechris-Pestov-Todorcevic correspondence: precompact expansions

In 2005, the paper "Fraiss\'e limits, Ramsey theory, and topological dynamics of automorphism groups" [KPT] by Kechris, Pestov and Todorcevic provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow. This immediately led to an explicit representation of this invariant in many concrete cases. However, in some particular situations, the framework of [KPT] does not allow to perform the computation directly, but only after a slight modification of the original argument. The purpose of the present paper is to supplement [KPT] in order to avoid that twist and to make it adapted for further applications.