Let $L$ be the set of sentences in some logic.  I am interested in cases where $L$ is the set of sentences in monadic second-order logic, or it is its $\Pi^1_1$ fragment.  Let $K$ be a class of finite structures.  The theory $\mathrm{Th}_L(K)$ of $K$ is the set of sentences $\phi$ such that $\forall M \in K$ $M \models \phi$.  I am interested in decidability of $\mathrm{Th}_L(K)$.For what kind of class $K$ is this known?  What are important / deep techniques here?  What is a good source to learn the results and the techniques?
Decidability of the monadic second-order theory of a class of finite structures
reference request;lo.logic;finite model theory