  Hydrodynamics provides a universal description of interacting quantum field
theories at sufficiently long times and wavelengths, but breaks down at scales
dependent on microscopic details of the theory. In the vicinity of a quantum
critical point, it is expected that some aspects of the dynamics are universal
and dictated by properties of the critical point. We use gauge-gravity duality
to investigate the breakdown of diffusive hydrodynamics in two low temperature
states dual to black holes with AdS$_2$ horizons, which exhibit quantum
critical dynamics with an emergent scaling symmetry in time. We find that the
breakdown is characterized by a collision between the diffusive pole of the
retarded Green's function with a pole associated to the AdS$_2$ region of the
geometry, such that the local equilibration time is set by infra-red properties
of the theory. The absolute values of the frequency and wavevector at the
collision ($\omega_{eq}$ and $k_{eq}$) provide a natural characterization of
all the low temperature diffusivities $D$ of the states via
$D=\omega_{eq}/k_{eq}^2$ where $\omega_{eq}=2\pi\Delta T$ is set by the
temperature $T$ and the scaling dimension $\Delta$ of an operator of the
infra-red quantum critical theory. We confirm that these relations are also
satisfied in an SYK chain model in the limit of strong interactions. Our work
paves the way towards a deeper understanding of transport in quantum critical
phases.
