Ordered materials passively self-assembled from dispersions of nanoparticles with steady interactions are subject to thermodynamic constraints on their phase separation kinetics forcing a trade-off between throughput and quality. Dynamically self-assembling dispersions whose interactions vary in a controlled way with time do not have these constraints and can rapidly form ordered structures while avoiding kinetic arrest. These out-of-equilibrium processes cannot be understood in terms of equilibrium thermodynamics or kinetic models derived from equilibrium thermodynamics, so new theories must be developed before dynamic self-assembly can be used to reliably fabricate nanomaterials. Here, we use dynamic simulation and theory to study the self-assembly kinetics of a monodisperse suspension of spherical nanoparticles interacting with a short-ranged, isotropic attraction that is toggled on and off cyclically in time. The rate of phase separation, local and global quality of the self-assembled structures, and range of tunable parameters leading to acceptable self-assembly are all enhanced with toggled attractions compared to steady attractions. The kinetic mechanism and rate of assembly can be easily controlled with the temporal toggling parameters. We develop simple phenomenological expressions to describe and predict the self-assembly rates for two predominant kinetic mechanisms. The first model describes the coarsening of percolated, gel-like networks, and the second describes the nucleation and growth of dense phases.