In the paper graphs of finite valency d are considered which admit a vertex primitive group of automorphisms. Theorem 1. Let \(d,r\in {\mathbb{N}}\) and \(f:{\mathbb{N}}\cup \{0\}\to {\mathbb{N}}\cup \{0\}\) such that \(f(n)=o(n)\) as \(n\to \infty.\) There exists a natural number c(d,f,r)\(\geq 1\) such that if G is a primitive group of automorphisms of a graph \(\Gamma\) of valency d, \(x\in V(\Gamma)\) and \(h\in G,\) h moves a vertex in \(B_{\Gamma}(x,r)\) and \(d_{\Gamma}(y,h(y))\leq f(d_{\Gamma}(x,y))\) for all \(y\in B_{\Gamma}(x,c(d,f,r)),\) then either the diameter of \(\Gamma\) does not exceed c(d,f,r) or the normal closure \(\) is an Abelian group. Here \(d_{\Gamma}\) denotes the graph theoretic distance in \(\Gamma\) and \(B_{\Gamma}(y,s)=\{z| z\in V(\Gamma)\) and \(d_{\Gamma}(y,z)\leq s\}\) for any vertex \(y\in V(\Gamma)\) and \(s\in {\mathbb{N}}.\) The proof makes use of the author's results on locally finite graphs. Using the proof of Sim's conjecture by Cameron-Praeger-Saxl-Seitz a consequence for finite graphs is obtained (Theorem 2). Also an interesting local analogue of Theorem 1 is obtained (Theorem 3). In the paper the author substantially clarifies his idea that it can turn out useful to study infinite graphs which are limits of finite graphs admitting primitive groups of automorphisms in order to study the (asymptotic) properties of finite primitive groups of permutations.