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In a recent paper [1] a new generalization of the Killing motion, the {\itgauged motion}, has been introduced for stationary spacetimes where it wasshown that the physical symmetries of such spacetimes are well describedthrough this new symmetry. In this article after a more detailed study in thestationary case we present the definition of gauged motion for generalspacetimes. The definition is based on the gauged Lie derivative induced by athreading family of observers and the relevant reparametrization invariance. Wealso extend the gauged motion to the case of Kaluza-Klein theories.Comment: 42 pages, revised version, typos correction along with some minor  changes, Revtex forma

Gauged motion in general relativity and in Kaluza-Klein theories