I will introduce a new method for obtaining the precise late-time asymptotic profile of solutions to geometric wave equations with inverse-square potentials on asymptotically flat spacetimes. This setting serves as a convenient toy model for understanding novel dynamical properties in the context of Einstein's equations of general relativity that arise in a variety of situations, e.g. when considering the gravitational properties of electromagnetically charged matter, when describing dynamical, rapidly rotating black holes and when considering higher, odd, spacetime dimensions.