In this conference, I have talked about two scenarios in which the out-of-equilibrium production of dark matter (DM) particles in the early universe is unavoidable. In the first one \cite{bhattacharyya_freezing-dark_2018}, we extend the standard model (SM) of particle physics by an extra $U(1)$ gauge group under which all the SM particles are neutral. We then consider DM candidates interacting only with the new spin-1 gauge boson, a heavy $Z^\prime$. We assume the presence of heavy beyond the SM fermions charged under both extra $U(1)$ and SM $SU(3)_c$, allowing for a feeble connection between DM and gluons. In the second scenario \cite{bernal_spin-2_2018}, we assume that the interaction between DM and SM particles are only mediated by gravitons and massive spin-2 fields, being therefore suppressed by the Planck and some intermediate scales, respectively. In both models, we show that the SM particles are able to produce the right amount of DM candidates via freeze-in at most in the early stages of the radiation era, for DM mass in the range $10^{-3}-10^{14}$ GeV. We have shown that if heavy mediators were produced on-shell within a period of entropy production in the early universe, as in the post-inflationary reheating, the DM relic density may be enhanced by many orders of magnitude relative to the usual instantaneous reheating approximation.