This talk is the last in a series of three presentations (together with Primo\v{z} Poto\v{c}nik and Pablo Spiga) concerning the Weiss conjecture and its generalisations. Let $\Gamma$ be a connected $G$-vertex-transitive graph, $v$ be a vertex of $\Gamma$ and $L$ be the permutation group induced by the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$ of $v$. The pair $(\Gamma,G)$ is said to be {\em locally}-$L$. A transitive permutation group $L$ is called \emph{graph-restrictive} if there exists a constant $c(L)$ such that, for every locally-$L$ pair $(\Gamma,G)$ and an arc $(u,v)$ of $\Gamma$, the arc stabiliser $G_{u,v}$ has order at most $c(L)$. We will discuss one of our new results, which allows us to prove graph-restrictiveness of certain permutation groups with previously unknown status. We will also discuss the status of the classification of transitive permutation groups of small degree according to graph-restrictiveness. In the second part of this talk, we discuss what can still be said in the case when $L$ is not graph-restrictive and consider growth of $|G_{u,v}|$ as a function of the order of $\Gamma$. Of particular interest to us is the case of 4-valent arc-transitive graphs (which turns out to be related to the 3-valent vertex-transitive case), about which we prove quite tight bounds.