Reionization in the early Universe is likely driven by dwarf galaxies. Using cosmological radiation-hydrodynamic simulations, we study star formation and the escape of Lyman continuum (LyC) photons from mini-haloes with $M$ $\lesssim$ 10$\mathrm{<mrow\; data-mjx-texclass="ORD">8</mrow>}$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$. Our simulations include a new thermo-turbulent star formation model, non-equilibrium chemistry and relevant stellar feedback processes (photoionization by young massive stars, radiation pressure and mechanical supernova explosions). We find that feedback reduces star formation very efficiently in mini-haloes, resulting in the stellar mass consistent with the slope and normalization reported in Kimm & Cen and the empirical stellar mass-to-halo mass relation derived in the local Universe. Because star formation is stochastic and dominated by a few gas clumps, the escape fraction in mini-haloes is generally determined by radiation feedback (heating due to photoionization), rather than supernova explosions. We also find that the photon number-weighted mean escape fraction in mini-haloes is higher (~ 20–40 per cent) than that in atomic-cooling haloes, although the instantaneous fraction in individual haloes varies significantly. The escape fraction from Pop III stars is found to be significant ( $\gtrsim$ 10 per cent) only when the mass is greater than ~100 M$\odot$. Based on simple analytic calculations, we show that LyC photons from mini-haloes are, despite their high escape fractions, of minor importance for reionization due to inefficient star formation. We confirm previous claims that stars in atomic-cooling haloes with masses 10$\mathrm{<mrow\; data-mjx-texclass="ORD">8</mrow>}$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$ $\lesssim$ $M$ $\lesssim$ 10$\mathrm{<mrow\; data-mjx-texclass="ORD">11</mrow>}$ M$\mathrm{<mrow\; data-mjx-texclass="ORD">\odot </mrow>}$ are likely to be the most important source of reionization.