  Models of stepped dark radiation have recently been found to have an
important impact on the anisotropies of the cosmic microwave background, aiding
in easing the Hubble tension. In this work, we study models with a sector of
dark radiation with a step in its abundance, which thermalizes after big bang
nucleosynthesis by mixing with the standard model neutrinos. For this, we
extend an earlier work which has focused on the background evolution only until
the dark sector thermalizes by deriving the full background and perturbation
equations of the model and implementing them in an Einstein-Boltzmann solving
code. We expound on the behavior of this model, discussing the wide range of
parameters that result in interesting and viable cosmologies that dynamically
generate dark radiation during a range of epochs. We find that for the strongly
self-coupled regime, there is no large cosmological impact for a tight prior on
the mass, whereas larger mass ranges allow a smooth interpolation between a
behavior close to the $\Lambda$CDM cosmological standard model and close to an
additional component of strongly self-interacting dark radiation. In the weakly
self-coupled regime we find that we can accommodate a parameter space relevant
for the neutrino anomalies as well as one relevant to easing the Hubble
tension.
