We use seismic ambient noise correlation and coda wave interferometry to estimate velocity variations at high temporal resolution, during the pre-eruptive period and the onset of the 2018 eruption of Kilauea volcano. A progressive velocity increase is observed from March to the end of April. It is followed by rapid decrease starting a few days before the onset of the East Rift Zone (ERZ) eruption and then by sharp velocity drop when the eruption started. The change of trend from velocity increase to decrease is progressively delayed by a few days from the summit caldera toward the ERZ. The location of the velocity perturbations shows a migration of the sources of velocity changes from the summit caldera toward the ERZ before the eruption. Using a model of pressure source, we show that the simultaneous caldera inflation and velocity increase probably result from an anisotropic distribution of fault and crack orientations. The velocity decrease could be due to damaging processes above the shallow reservoir and to plastic deformations around the caldera. We introduce a forward model of rock damage associated with the volcano-tectonic seismicity to calculate the velocity decrease. The good agreement between the calculated and the observed velocity variations shows that a large part of the velocity decrease results from damage of the medium. The delayed onsets of velocity decrease and the source migration of velocity perturbations are probably related to progressive fault openings in the Southern and Eastern parts of the caldera and to magma transfer toward the ERZ.