  Reionisation 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 $M_{\rm halo}
\le 10^8\,M_\odot$. Our simulations include a new thermo-turbulent star
formation model, non-equilibrium chemistry, and relevant stellar feedback
processes (photoionisation 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 normalisation 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
photo-ionisation), rather than supernova explosions. We also find that the
photon number-weighted mean escape fraction in mini-haloes is higher
($\sim20$-$40\%$) 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 ($\ge10\%$) only when
the mass is greater than $\sim$100\,\msun. Based on simple analytic
calculations, we show that LyC photons from mini-haloes are, despite their high
escape fractions, of minor importance for reionisation due to inefficient star
formation. We confirm previous claims that stars in atomic-cooling haloes with
masses $10^8\,M_\odot\le M_{\rm halo} \le 10^{11}\,M_\odot$ are likely to be
the most important source of reionisation.
