  Onset of the instability of a multiple-scattering speckle pattern in a random
medium with Kerr nonlinearity is significantly affected by the noninstantaneous
character of the nonlinear medium response. The fundamental time scale of the
spontaneous speckle dynamics beyond the instability threshold is set by the
largest of times $T_{\mathrm{D}}$ and $\tau_{\mathrm{NL}}$, where
$T_{\mathrm{D}}$ is the time required for the multiple-scattered waves to
propagate through the random sample and $\tau_{\mathrm{NL}}$ is the relaxation
time of the nonlinearity. Inertial nature of the nonlinearity should complicate
the experimental observation of the instability phenomenon.
