An $(n_{k})$ configuration is a collection of points and straight lines, usually in the Euclidean plane, so that each point lies on $k$ lines and each line passes through $k$ points; such a configuration will be called symmetric if it possesses non-trivial geometric symmetry. Although examples of symmetric $(n_{3})$ configurations with continuous parameters are known, to this point, all known connected infinite families of $(n_{4})$ configurations with non-trivial geometric symmetry had the property that each set of discrete parameters describing the configuration corresponded to a single $(n_{4})$ configuration. This paper presents several new classes of highly symmetric $(n_{4})$ configurations which have at least one continuous parameter; that is, the configurations are movable.