This work presents the modelling fundamentals to study the dynamics of inverter-based resources (IBRs) in weak distribution grids and derive disturbance performance characteristics describing how they should behave under different conditions. More specifically, with respect to small-signal disturbances we study the possible voltage violations following frequency response from IBRs due to low system strength and also propose specific design requirements for IBR d-q current control to guarantee a stable response. In the context of large-signal disturbances, we highlight how active power-voltage control may not be effective due to a delay imposed by the physical features of the distribution network. Further, we mathematically discuss how IBR reactive power control could enhance its phase-locked loop stability. The proposed disturbance performance characteristics are then integrated into the IBR converter control via a novel voltage-priority reference generation strategy. Simulation results on a real Australian network show the efficacy of the proposed operational and control design requirements, and highlight possible unexpected active/reactive power interactions in weak distribution grids.