<p>The local equilibration time of quantum many-body systems has been conjectured to satisfy a Planckian bound, so that it always exceeds some value on the order of <i>ℏ</i>/<i>T</i>, where <i>T</i> is the temperature of the system. Here we provide a sharp and universal definition of the local equilibration timescale, and show that it is bounded below by the strong-coupling scale of diffusive fluctuations, which can be expressed in terms of familiar transport parameters. When applied to conformal field theories at a finite temperature, this result produces the Planckian bound. Moreover, this fluctuation bound applies to any local thermalizing system. We study its implication for correlated insulators, metals and disordered fixed points, where it can be used to establish a lower bound on diffusivity in terms of specific heat. Finally, we discuss how the local equilibration time can be directly measured in experiments.</p>