Excess emission, associated with warm, dust belts, commonly known as exozodis, has been observed around a third of nearby stars. The high levels of dust required to explain the observations are not generally consistent with steady-state evolution. A common suggestion is that the dust results from the aftermath of a dynamical instability, an event akin to the Solar System's Late Heavy Bombardment. In this work, we use a database of N-body simulations to investigate the aftermath of dynamical instabilities between giant planets in systems with outer planetesimal belts. We find that, whilst there is a significant increase in the mass of material scattered into the inner regions of the planetary system following an instability, this is a short-lived effect. Using the maximum lifetime of this material, we determine that even if every star has a planetary system that goes unstable, there is a very low probability that we observe more than a maximum of 1% of sun-like stars in the aftermath of an instability, and that the fraction of planetary systems currently in the aftermath of an instability is more likely to be limited to