We use state-of-art measurements of the galaxy luminosity function (LF) at z=6, 7 and 8 to derive constraints on warm dark matter (WDM), late-forming dark matter (LFDM) and ultra-light axion dark matter (ULADM) models alternative to the cold dark matter (CDM) paradigm. To this purpose we have run a suite of high-resolution N-body simulations to accurately characterise the low mass-end of the halo mass function and derive DM model predictions of the high-z luminosity function. In order to convert halo masses into UV-magnitudes we introduce an empirical approach based on halo abundance matching which allows us to model the LF in terms of the amplitude and scatter of the ensemble average star formation rate halo mass relation of each DM model, $langle {rm SFR}({rm M_{ h}},z)rangle$. We find that independent of the DM scenario the average SFR at fixed halo mass increases from z=6 to 8, while the scatter remains constant. At halo mass ${rm M_{h}}gtrsim 10^{12},{rm M}_odot$ h$^{-1}$ the average SFR as function of halo mass follows a double power law trend that is common to all models, while differences occur at smaller masses. In particular, we find that models with a suppressed low-mass halo abundance exhibit higher SFR compared to the CDM results. Using deviance statistics we obtain a lower limit on the WDM thermal relic particle mass, $m_{rm WDM}gtrsim 1.5$ keV at $2sigma$. In the case of LFDM models, the phase transition redshift parameter is bounded to $z_tgtrsim 8cdot 10^5$ at $2sigma$. We find ULADM best-fit models with axion mass $m_agtrsim 1.6cdot 10^{-22}$ eV to be well within $2sigma$ of the deviance statistics. We remark that measurements at $z=6$ slightly favour a flattening of the LF at faint UV-magnitudes. This tends to prefer some of the non-CDM models in our simulation suite, although not at a statistically significant level to distinguish them from CDM.
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