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Background: $\ensuremath{\beta}$-decay rates of neutron-rich nuclei are a crucial ingredient in simulations of the $r$-process nucleosynthesis. Up to now global calculations of these rates have been performed within the quasiparticle random-phase approximation (QRPA), which is known to have limited accuracy and predictive power due to a limited treatment of nuclear correlations. Although extensions of this approach have been developed to include a more precise description of correlations, and have been applied to the study of allowed $\ensuremath{\beta}$-decay of selected nuclei, no systematic calculations including first-forbidden transitions have so far been performed.Purpose: The goal of this work is to compute $\ensuremath{\beta}$-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 to study the role of first-forbidden transitions in a framework that includes complex nucleonic correlations beyond the QRPA.Method: The many-body approach based on the relativistic QRPA extended to account for the coupling between single-particle and collective degrees of freedom is applied. Both Gamow-Teller and first-forbidden contributions to the decay rates are considered.Results: Overall, the fragmentation of the transition strength distributions due to the coupling between single particles and collective vibrations systematically increases the transition strength at low excitation energies and yields a decrease of the $\ensuremath{\beta}$-decay half-lives which is particularly important for low $Q$ values. Such effects are crucial to bring Gamow-Teller transitions within the decay energy window, and to reproduce the measured decay rates. Generally, quasiparticle-vibration coupling tends to decrease the probability of decay via FF transition near stability, due to the appearance of Gamow-Teller transitions within the decay $Q$ value. While in 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 ${1}^{\ensuremath{-}}$ and ${0}^{\ensuremath{-}}$ operators.Conclusions: The relativistic approach based on the coupling between nucleons and vibrations has the ability to capture relevant correlations that are essential for an accurate predictions of $\ensuremath{\beta}$ decay and provides an ideal framework for future large-scale calculations. Upcoming experimental measurements of $\ensuremath{\beta}$-decay rates in the $N=126$ region by radioactive-beam facilities will be crucial in order to validate the approach.