Liquid droplets of biomolecules play key roles in organizing cellular behavior, and are also technologically relevant, yet physical studies of dynamic processes of such droplets have generally been lacking. Here, we investigate and quantify the dynamics of formation of dilute internal inclusions, i.e., vacuoles, within a model system consisting of liquid droplets of DNA ‘nanostar’ particles. When acted upon by DNA-cleaving restriction enzymes, these DNA droplets exhibit cycles of appearance, growth, and bursting of internal vacuoles. Analysis of vacuole growth shows their radius increases linearly in time. Further, vacuoles pop upon reaching the droplet interface, leading to droplet motion driven by the osmotic pressure of restriction fragments captured in the vacuole. We develop a model that accounts for the linear nature of vacuole growth, and the pressures associated with motility, by describing the dynamics of diffusing restriction fragments. The results illustrate the complex non-equilibrium dynamics possible in biomolecular condensates.