Let 𝒦 be a class of finite relational structures. We define ℰ𝒦 to be the class of finite relational structures A such that A/E ∈ 𝒦, where E is an equivalence relation defined on the structure A. Adding arbitrary linear orderings to structures from ℰ𝒦, we get the class 𝒪ℰ𝒦. If we add linear orderings to structures from ℰ𝒦 such that each E-equivalence class is an interval then we get the class 𝒞ℰ[𝒦*]. We provide a list of Fraïssé classes among ℰ𝒦, 𝒪ℰ𝒦 and 𝒞ℰ[𝒦*]. In addition, we classify 𝒪ℰ𝒦 and 𝒞ℰ[𝒦*] according to the Ramsey property. We also conduct the same analysis after adding additional structure to each equivalence class. As an application, we give a topological interpretation using the technique introduced in Kechris, Pestov and Todorčević. In particular, we extend the lists of known extremely amenable groups and universal minimal flows.