The use of differential forms (and exterior calculus) to represent physical quantities enables a dimension and coordinate system independent description of continuum mechanics valid on arbitrary manifolds. It also forms the basis for a powerful approach to building numerical models: structure-preserving discretizations. These schemes can obtain discrete analogues of key properties such as conservation laws and freedom from spurious/unphysical computational modes. Common approaches include compatible Galerkin methods and discrete exterior calculus (DEC). However, in addition to such structural properties, continuum mechanics models also need a good representation of transport (i.e. advection). Previous work has shown how to obtain structure-preserving, high-order, oscillation-limiting (SPHOOL) DEC transport operators for n-forms (densities) when using a velocity-based representation, through the inherent topological/metric splitting that arises through an exterior calculus formulation. In the talk we will discuss recent efforts to extend these ideas to arbitrary k-forms and both velocity and momentum-based representations.