Dark matter (DM) from freeze-in or superWIMP production is well known to imprint non-cold DM signatures on cosmological observables. We derive constraints from Lyman-$\alpha$ forest observations for both cases, basing ourselves on a reinterpretation of the existing Lyman-$\alpha$ limits on thermal warm DM. We exclude DM masses below 15 keV for freeze-in, in good agreement with previous literature, and provide a generic lower mass bound for superWIMPs that depends on the mother particle decay width. Special emphasis is placed on the mixed scenario, where contributions from both freeze-in and superWIMP are similarly important. In this case, the imprint on cosmological observables can deviate significantly from thermal warm DM. Furthermore, we provide a modified version of the Boltzmann code CLASS, analytic expressions for the DM distributions, and fits to the DM transfer functions that account for both mechanisms of production. Moreover, we also derive generic constraints from $\Delta N_\mathrm{eff}$ measurements and show that they cannot compete with those arising from Lyman-$\alpha$ observations. For illustration, we apply the above generic limits to a coloured $t$-channel mediator DM model, in which case contributions from both freeze-in through scatterings and decays, as well as superWIMP production can be important. We map out the entire cosmologically viable parameter space, cornered by bounds from Lyman-$\alpha$ observations, the LHC, and Big Bang Nucleosynthesis., Comment: 31 pages + references, 5 figures, v2: Legend of Fig. 2 corrected, Appendix A expanded, references added