  We introduce a general class of transport distances ${\rm WB}_{\Lambda}$ over
the space of positive semi-definite matrix-valued Radon measures
$\mathcal{M}(\Omega,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures
distance. Such a distance is defined via a generalized Benamou-Brenier
formulation with a weighted action functional and an abstract matricial
continuity equation, which leads to a convex optimization problem. Some
recently proposed models, including the Kantorovich-Bures distance and the
Wasserstein-Fisher-Rao distance, can naturally fit into ours. We give a
complete characterization of the minimizer and explore the topological and
geometrical properties of the space $(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm
WB}_{\Lambda})$. In particular, we show that
$(\mathcal{M}(\Omega,\mathbb{S}_+^n),{\rm WB}_{\Lambda})$ is a complete
geodesic space and exhibits a conic structure.
