One of the most prominent features of quantum entanglement is its invariability under local unitary transformations, which implies that the degree of entanglement or nonseparability remains constant during free-space propagation, true for both quantum and classically entangled modes. Here we demonstrate an exception to this rule using a carefully engineered vectorial light field, and we study its nonseparability dynamics upon free-space propagation. We show that the local nonseparability between the spatial and polarization degrees of freedom dramatically decays to zero while preserving the purity of the state and hence the global nonseparability. We show this by numerical simulations and corroborate it experimentally. Our results evince novel properties of classically entangled modes and point to the need for new measures of nonseparability for such vectorial fields, while paving the way for novel applications for customized structured light.