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The Froggatt-Nielsen (FN) mechanism provides an attractive way of generating the determined fermion mass hierarchy and quark mixing matrix elements in the Standard Model (SM). Here we extend it by coupling the FN field, the flavon, to a dark sector containing one or more dark matter particles which are produced non-thermally sequentially through flavon production. Non-thermal flavon production occurs efficiently via freeze-in and through field oscillations. We explore this in the regime of high-scale breaking $Λ$ of the global $U(1)_{\textrm{FN}}$ group and at the reheating temperature $T_R\ll Λ$ where the flavon remains out of equilibrium at all times. We identify phenomenologically acceptable regions of $T_R$ and the flavon mass where the relic abundance of dark matter and other cosmological constraints are satisfied. In the case of one-component dark matter we find an effective upper limit on the FN charges at high $Λ$, i.e. $Q_{\rm FN}^{\rm DM}\leq13$. In the multi-component dark sector scenario the dark matter can be the heaviest dark particle that can be effectively stable at cosmological timescales, alternatively it can be produced sequentially by decays of the heavier ones. For scenarios where dark decays occur at intermediate timescales, i.e. $t\sim 0.1- 10^{28}\,{\rm s}$, we find that existing searches can effectively probe interesting regions of parameter space. These searches include indirect probes on decays such as $γ$-ray and neutrino telescopes as well as analyses of the Cosmic Microwave Background, as well as constraints on small scale structure formation from the Lyman-$α$ forest. We comment on the future prospects of such probes, place projected sensitivities, and discuss how this scenario could accommodate the cosmological $S_8$ tension.