  Tensions between cosmological parameters (in particular the local expansion
rate $H_0$ and the amplitude of matter clustering $S_8$) inferred from
low-redshift data and data from the cosmic microwave background (CMB) and
large-scale structure (LSS) experiments have inspired many extensions to the
standard cosmological model, $\Lambda$CDM. Models which simultaneously lessen
both tensions are of particular interest. We consider one scenario with the
potential for such a resolution, in which some fraction of the dark matter has
converted into dark radiation since the release of the CMB. Such a scenario
encompasses and generalizes the more standard "decaying dark matter" model,
allowing additional flexibility in the rate and time at which the dark matter
converts into dark radiation. In this paper, we constrain this scenario with a
focus on exploring whether it can solve (or reduce) these tensions. We find
that such a model is effectively ruled out by low-$\ell$ CMB data, in
particular by the reduced peak-smearing due to CMB lensing and the excess
Integrated Sachs--Wolfe (ISW) signal caused by the additional dark energy
density required to preserve flatness after dark matter conversion into dark
radiation. Thus, such a model does not have the power to reduce these tensions
without further modifications. This conclusion extends and generalizes related
conclusions derived for the standard decaying dark matter model.
