While cold dark matter numerical simulations predict steep, `cuspy' density profiles for dark matter halos, observations favour shallower `cores'. The introduction of baryonic physics alleviates this discrepancy, notably as feedback-driven outflow episodes contribute to expand the dark matter distribution. But complex hydrodynamical simulations do not necessarily specify the physical mechanisms through which baryons affect the dark matter distribution. I will present two simple theoretical models describing core formation. In the first one, sudden bulk outflows induced by stellar feedback reorganise the halo mass distribution while it relaxes to a new equilibrium. In the second one, small stochastic density fluctuations induce kicks to collisionless particles that progressively deviate them from their orbits. Both models are tested against numerical simulations and provide a simple understanding of the transition from cusps to cores by feedback-driven outflows. Since the process at stake during dark matter core formation can also puff up the stellar distribution, it may further explain the formation of ultra-diffuse galaxies.