ABSTRACT We present a simple model for the response of a dissipationless spherical system to an instantaneous mass change at its centre, describing the formation of flat cores in dark matter haloes and ultra-diffuse galaxies (UDGs) from feedback-driven outflow episodes in a specific mass range. This model generalizes an earlier simplified analysis of an isolated shell into a system with continuous density, velocity, and potential profiles. The response is divided into an instantaneous change of potential at constant velocities due to a given mass-loss or mass-gain, followed by energy-conserving relaxation to a new Jeans equilibrium. The halo profile is modelled by a two-parameter function with a variable inner slope and an analytic potential profile, which enables determining the associated kinetic energy at equilibrium. The model is tested against NIHAO cosmological zoom-in simulations, where it successfully predicts the evolution of the inner dark matter profile between successive snapshots in about 75 per cent of the cases, failing mainly in merger situations. This model provides a simple understanding of the formation of dark matter halo cores and UDGs by supernova-driven outflows, and a useful analytic tool for studying such processes.