  We report a fully microscopic theory for transconductivity, or, equivalently,
momentum transfer rate, of Coulomb coupled electron systems. We use the Kubo
linear response formalism, and our main formal result expresses the
transconductivity in terms of two fluctuation diagrams, which are topologically
related, but not equivalent to, the Aslamazov-Larkin diagrams known for
superconductivity. Previously reported results are shown to be special cases of
our general expression; specifically, for constant impurity scattering rates,
we recover the Boltzmann equation results in the semiclassical clean limit, and
the memory function results in dirty systems. Furthermore, we show that for
energy dependent relaxation times, the final result is not expressible in terms
of standard density-response functions. Other new results include: (i) at T=0,
the frequency dependence of the transfer rate is found to be proportional to
$\Omega$ and $\Omega^2$ for frequencies below and above the impurity scattering
rate, respectively and (ii) the weak localization correction to the
transconductivity is given by $\delta\sigma^{WL}_{21} \propto
\delta\sigma^{WL}_{11} +\delta\sigma^{WL}_{22}$.
