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Low-mass galaxies are thought to provide the bulk of the ionizing radiation necessary to reionize the Universe. The amount of photons escaping the galaxies is poorly constrained theoretically, and difficult to measure observationally. Yet it is an essential parameter of reionization models. We study in detail how ionizing radiation can leak from high-redshift galaxies. For this purpose, we use a series of high-resolution radiation hydrodynamics simulations, zooming on three dwarf galaxies in a cosmological context. We find that the energy and momentum input from the supernova explosions has a pivotal role in regulating the escape fraction by disrupting dense star-forming clumps, and clearing sightlines in the halo. In the absence of supernovae, photons are absorbed very locally, within the birth clouds of massive stars. We follow the time evolution of the escape fraction and find that it can vary by more than six orders of magnitude. This explains the large scatter in the value of the escape fraction found by previous studies. This fast variability also impacts the observability of the sources of reionization: a survey even as deep as M1500 = -14 would miss about half of the underlying population of Lyman-continuum emitters. [ABSTRACT FROM AUTHOR]