In this talk, I will describe how Riemannian submersions on a spacetime of the form $M_4 \times  K$ relate to $\mathrm{Diff}(K)$-gauge theories on the base manifold $M_4$. This framework generalizes the usual Kaluza–Klein ansatz by allowing the fibres of the submersion to have variable geometry and not be totally geodesic. In this case, the higher-dimensional metric encodes both massless and massive 4D gauge fields, as well as a non-trivial Higgs sector. I will discuss new features of these geometrical models and their implications for 4D physics. For example, how geodesics in higher-dimensions correctly indicate that the classical mass and charge of a test particle, as perceived in 4D, can change in regions of spacetime where massive gauge fields are present.