  We study the late-time relaxation following a quench in a driven-dissipative
quantum many-body system. We consider the open Dicke model, describing the
infinite-range interactions between $N$ atoms and a single, lossy
electromagnetic mode. We show that the dynamical phase transition at a critical
atom-light coupling is characterized by the interplay between reservoir-driven
and intrinsic relaxation processes. Above the critical coupling, small
fluctuations in the occupation of the dominant quasiparticle-mode start to grow
in time while the quasiparticle lifetime remains finite due to losses. Near the
critical interaction strength we observe a crossover between exponential and
power-law $1/\tau$ relaxation, the latter driven by collisions between
quasiparticles. For a quench exactly to the critical coupling, the power-law
relaxation extends to infinite times, but the finite lifetime of quasiparticles
prevents ageing to appear. We predict our results to be accessible to quench
experiments with ultracold bosons in optical resonators.
