  We determine the structure of equilibrium states for a natural dynamics on
the boundary quotient diagram of $C^*$-algebras for a large class of right LCM
semigroups. The approach is based on abstract properties of the semigroup and
covers the previous case studies on $\mathbb{N} \rtimes \mathbb{N}^\times$,
dilation matrices, self-similar actions, and Baumslag-Solitar monoids. At the
same time, it provides new results for large classes of right LCM semigroups,
including those associated to algebraic dynamical systems.
