  We study experimentally observable signals for nonlinear QCD dynamics in deep
inelastic scattering (DIS) at small Bjorken variable $x$ and moderate
virtuality $Q^2$, by quantifying differences between the linear
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution and nonlinear
evolution with the Balitsky-Kovchegov (BK) equation. To remove the effect of
the parametrization freedom in the initial conditions of both equations, we
first match the predictions for the DIS structure functions $F_2$ and $F_{\rm
L}$ from both frameworks in a region in $x,Q^2$ where both frameworks should
provide an accurate description of the relevant physics. The differences in the
dynamics are then quantified by the deviations when one moves away from this
matching region. For free protons we find that the differences in $F_2$ remain
at a few-percent level, while in $F_{\rm L}$ the deviations are larger, up to
$10\,\%$ at the EIC and $40\,\%$ at the LHeC kinematics. With a heavy nucleus
the differences are up to $10\,\%$ in $F_2$, and can reach $20\,\%$ and
$60\,\%$ in $F_{\rm L}$ for the EIC and the LHeC, respectively.
