  $\beta$-decay rates of neutron-rich nuclei, in particular those located at
neutron shell closures, play a central role in simulations of the heavy-element
nucleosynthesis and resulting abundance distributions. We present $\beta$-decay
half-lives of even-even $N=82$ and $N=126$ $r$-process waiting-point nuclei
calculated in the approach based on relativistic quasiparticle random phase
approximation with quasiparticle-vibration coupling. The calculations include
both allowed and first-forbidden transitions. In the $N=82$ chain, the
quasiparticle-vibration coupling has an important impact close to stability, as
it increases the contribution of Gamow-Teller modes and improves the agreement
with the available data. In the $N=126$ chain, we find the decay to proceed
dominantly via first-forbidden transitions, even when the coupling to
vibrations is included.
