  We develop a Boltzmann-Langevin description of Coulomb drag effect in clean
double-layer systems with large interlayer separation $d$ as compared to the
average interelectron distance $\lambda_F$. Coulomb drag arises from density
fluctuations with spatial scales of order $d$. At low temperatures, their
characteristic frequencies exceed the intralayer equilibration rate of the
electron liquid, and Coulomb drag may be treated in the collisionless
approximation. As temperature is raised, the electron mean free path becomes
short due to electron-electron scattering. This leads to local equilibration of
electron liquid, and consequently drag is determined by hydrodynamic density
modes. Our theory applies to both the collisionless and the hydrodynamic
regimes, and it enables us to describe the crossover between them. We find that
drag resistivity exhibits a nonmonotonic temperature dependence with multiple
crossovers at distinct energy scales. At the lowest temperatures, Coulomb drag
is dominated by the particle-hole continuum, whereas at higher temperatures of
the collision-dominated regime it is governed by the plasmon modes. We observe
that fast intralayer equilibration mediated by electron-electron collisions
ultimately renders a stronger drag effect.
