The equations of motion are derived here with the help of the Heisenberg EoM. Starting in the Heisenberg picture the time evolution of an operator is \begin{equation} \frac{\partial}{\partial t} \hat{O}_\text{H} (x,t) = \left[ \hat{O}_\text{H} (x,t) , \hat{H}_\text{H} (t) \right] + \left( \frac{\partial}{\partial t} \hat{O}_\text{S} (x) \right)_\text{H} \quad . \end{equation} Under the following conditions, the EoM for the electron current $J(x,t)$ becomes \begin{equation} \frac{\partial}{\partial t} J (x,t) = \dots \quad . \end{equation} This holds for symmetric correlation potentials \begin{equation} \nu (\vec{r},\vec{r}^\prime) = \nu (\vec{r}^\prime,\vec{r}) \end{equation} and \begin{equation} \frac{\partial}{\partial t} \frac{\partial}{\partial k} = \frac{\partial}{\partial k} \frac{\partial}{\partial t} \quad . \end{equation} The calculation can be found here.