Cosmic Dawn (CoDa) II yields the first statistically-meaningful determination of the relative contribution to reionization by galaxies of different halo mass, from a fully-coupled radiation-hydrodynamics simulation of the epoch of reionization large enough ($sim$ 100 Mpc) to model global reionization while resolving the formation of all galactic halos above $sim 10^8 M_odot$. Cell transmission inside high-mass haloes is bi-modal -- ionized cells are transparent, while neutral cells absorb the photons their stars produce - and the halo escape fraction $f_{esc}$ reflects the balance of star formation rate (SFR) between these modes. The latter is increasingly prevalent at higher halo mass, driving down $f_{esc}$ (we provide analytical fits to our results), whereas halo escape luminosity, proportional to $f_{esc} times$SFR, increases with mass. Haloes with dark matter masses within $6.10^{8} M_odot < M_h < 3.10^{10} M_odot$ produce $sim 80$% of the escaping photons at z=7, when the Universe is 50% ionized, making them the main drivers of cosmic reionization. Less massive haloes, though more numerous, have low SFRs and contribute less than 10% of the photon budget then, despite their high $f_{esc}$. High mass haloes are too few and too opaque, contributing $<10$% despite their high SFRs. The dominant mass range is lower (higher) at higher (lower) redshift, as mass function and reionization advance together (e.g. at z$=8.5$, x$_{rm HI}=0.9$, $M_h < 5.10^9 M_odot$ haloes contributed $sim$80%). Galaxies with UV magnitudes $M_{AB1600}$ between $-12$ and $-19$ dominated reionization between z$=6$ and 8.
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