We derive an expression for the drag rate (i.e., interlayer momentum transfer rate) for carriers in two coupled two-dimensional gases to lowest nonvanishing order in the screened interlayer electron-electron interaction, valid for arbitrary intralayer scattering mechanisms, using the Boltzmann transport equation. We calculate the drag rate for experimentally relevant parameters, and show that for moderately high temperatures (T≳0.2TF, where TF is the Fermi temperature) the dynamical screening of the interlayer results in a large enhancement of the drag rate due to the presence of coupled plasmon modes. This plasmon enhancement causes the scaled drag rate to have a peak (i) as a function of temperature at T≊0.5TF, and (ii) as a function of the ratio of densities of the carriers in the two layers when their Fermi velocities are equal. We also show that the drag rate can be significantly affected by the intralayer scattering mechanisms; in particular, the drag rate changes approximately by a factor of 2 when the dopant-layer modulation-doped structures are moved in from 400 to 100 Å.