  Non-thermalized dark matter is a cosmologically viable alternative to the
widely studied weakly interacting massive particle. We study the evolution of
the dark matter phase-space distributions arising from freeze-in and superWIMP
production as well as the combination of both. Utilizing our implementation in
CLASS, we investigate the cosmological imprints on the matter power spectrum,
constrained by Lyman-$\alpha$ forest observations. For the explicit example of
a colored $t$-channel mediator model, we explore the cosmologically allowed
parameter space highlighting the interplay of Lyman-$\alpha$ constraints with
those from Big Bang Nucleosynthesis and the LHC.
