  A generalized unbalanced optimal transport distance ${\rm WB}_{\Lambda}$ on
matrix-valued measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$ was defined in
[arXiv:2011.05845] \`{a} la Benamou-Brenier, which extends the
Kantorovich-Bures and the Wasserstein-Fisher-Rao distances. In this work, we
investigate the convergence properties of the discrete transport problems
associated with ${\rm WB}_{\Lambda}$. We first present a convergence framework
for abstract discretization. Then, we propose a specific discretization scheme
that aligns with this framework, under the assumption that the initial and
final distributions are absolutely continuous with respect to the Lebesgue
measure. Moreover, thanks to the static formulation, we show that such an
assumption can be removed for the Wasserstein-Fisher-Rao distance.
