Using normal quotient techniques developed by the first author, the main theorem applies to general G-vertex-transitive graphs which are G-locally primitive (respectively, G-locally quasiprimitive), that is, the stabiliser of a vertex 伪 acts primitively (respectively quasiprimitively) on the set of vertices adjacent to 伪. We discuss how our results may be used to investigate conjectures by Richard Weiss (in 1978) and the first author (in 1998) that the order of is bounded above by some function depending only on the valency of 螕, when 螕 is G-locally primitive or G-locally quasiprimitive, respectively.