  The '$S_8$ tension' is a longstanding discrepancy between the cosmological
and local determination of the amplitude of matter fluctuations, parameterized
as $S_8\equiv\sigma_8(\Omega_m/0.3)^{0.5}$, where $\sigma_8$ is the root mean
square of matter fluctuations on a 8 $h^{-1}$Mpc scale, and $\Omega_m$ is the
total matter abundance. It was recently shown that dark matter (DM) decaying
into a massless (dark radiation) and a massive (warm DM) species, with a
lifetime $\Gamma^{-1} \simeq 55~ (\varepsilon/0.007)^{1.4}$ Gyrs -- where
$\varepsilon$ represent the mass-energy fraction transferred to the massless
component -- can resolve the tension. Thanks to a new, fast and accurate
approximation scheme for the warm species, we perform a comprehensive study of
this 2-body decaying DM scenario, discussing in details its dynamics and its
impact on the CMB and linear matter power spectra. We then confront the
robustness of the resolution to the '$S_8$ tension' against a number of changes
in the analysis: different $S_8$ priors, marginalization over the lensing
information in Planck data, trading Planck high$-\ell$ polarization data for
those from the SPTpol collaboration, and the inclusion of the recent results
from the Xenon1T collaboration. We conclude that the preference for decaying
DM, while entirely driven by the local $S_8$ measurements, does not sensibly
degrade the fit to any of the cosmological data-sets considered, and that the
model could explain the anomalous electron recoil excess reported by the
Xenon1T collaboration.
