  We compute $\beta$-decay half-lives of isotonic nuclear chains located at
neutron shell closures $N=50$, $82$, $126$ and $184$, which are of particular
importance for the $r$-process nucleosynthesis, and study the role of
first-forbidden transitions in a framework that includes complex nucleonic
correlations beyond the quasiparticle random phase approximation. Such
correlations are accounted for by coupling single nucleons to collective
degrees of freedom (nuclear vibrations), and are found essential to reproduce
available experimental $\beta$-decay rates and more precise many-body methods.
We find that the nucleon-vibration correlations tend to decrease the
probability of decay via first-forbidden transition near stability, as they
enhance Gamow-Teller transitions at low energy. While in the lighter systems
allowed transitions dominate, the decay of $N=126$ and $N=184$ nuclei is found
to occur to a large extent via first-forbidden transitions, and in particular
those induced by $1^-$ and $0^-$ operators. Overall the many-body method based
on nucleon-vibration coupling provides an ideal framework for future
large-scale calculations. Upcoming experimental measurements of $\beta$-decay
rates in the $N=126$ region by radioactive-beam facilities will be crucial in
order to validate the approach.
