We study reparametrization-invariant systems, mainly the relativistic particle and its $D$-dimensional extended object generalization $d$-brane. The corresponding matter Lagrangians naturally contain background interactions, like electromagnetism and gravity. For a $d$-brane that doesn’t alter the background fields, we define non-relativistic equations assuming integral sub-manifold embedding of the $d$-brane. The mass-shell constraint and the Klein–Gordon equation are shown to be universal when gravity-like interaction is present. Our approach to the Dirac equation follows Rund’s technique for the algebra of the $\gamma$-matrices that doesn’t rely on the Klein–Gordon equation.