  We develop a hydrodynamic description for the driven-dissipative dynamics of
the entanglement negativity, which quantifies the genuine entanglement in
mixed-state systems. We focus on quantum quenches in fermionic and bosonic
systems subject to linear dissipation, as described by quadratic Lindblad
master equations. In the spirit of hydrodynamics, we divide the system into
mesoscopic cells. At early times, correlations are generated in each cell by
the unitary component of the evolution. Correlations are then transported
across different cells via ballistic quasiparticle propagation, while
simultaneously evolving under the action of the environment. We show that in
the hydrodynamic limit the negativity can be reconstructed from the
correlations between the independently propagating quasiparticles. We benchmark
our approach considering quenches from both homogeneous and inhomogeneous
initial states in the Kitaev chain, the tight-binding chain, and the harmonic
chain in the presence of gain/loss dissipation.
