  The local equilibration time of quantum many-body systems has been
conjectured to satisfy a "Planckian bound", $\tau_{\rm eq}\gtrsim
\frac{\hbar}{T}$. We provide a sharp and universal definition of this time
scale, 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 finite temperature,
this fluctuation bound implies the Planckian bound. The fluctuation bound
moreover applies to any local thermalizing system, and we study its implication
for correlated insulators, metals, as well as disordered fixed points, where it
can be used to establish a lower bound on the diffusivity in terms of the
specific heat $D\gtrsim 1/(c_V^{2/d}\tau_{\rm eq})$. Finally, we discuss how
the local equilibration time can be directly measured in experiments.
