If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang–Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an FC-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an FC-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element b of $G(X,r)$ there are finitely many elements of the form $b* c$ and $c* b$, with $cın G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of FC-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.