This paper presents a novel numerical model on unstructured grids for all-speed flows using the multimoment constrained finite volume method, where the point values at both the cell center and cell vertices are updated in time as the computational variables. These two kinds of point values are solved by different numerical procedures with different Riemann solvers, and then they are used as the local degrees of freedom to generate high-order reconstructions. Numerical oscillation is effectively suppressed by a multidimensional limiting projection. The numerical dissipation is controlled by the boundary variation diminishing principle that adaptively chooses the interpolation function so as to minimize the jumps (variations) across the cell boundaries. Numerical results show that the present solver has third-order accuracy for a smooth solution and can resolve both continuous and discontinuous solutions with competitive quality.