This paper considers a class of second-order accurate vertex centered mixed finite element finite-volume MUSCL schemes. These schemes apply to unstructured triangulations and tetrahedrizations and fluxes are computed on an edge basis. We define conditions under which these schemes satisfy a density-positivity statement for Euler flows, a maximum principle for a scalar conservation law and a multicomponent flow. This extends to an Arbitrary- Lagrangian-Eulerian formulation. Steady and unsteady flow simulations illustrate the accuracy and the robustness of these schemes.