  In this note we study the generation of $C_0$-semigroups by first order
differential operators on $\mathrm{L}^p (\mathbb{R}_+,\mathbb{C}^{\ell})\times
\mathrm{L}^p ([0,1],\mathbb{C}^{m})$ with general boundary conditions. In many
cases we are able to characterize the generation property in terms of the
invertibility of a matrix associated to the boundary conditions. The abstract
results are used to study well-posedness of transport equations on non-compact
metric graphs.
