  We find stochastic equations governing eigenvalues and eigenvectors of a
dynamical complex Ginibre ensemble reaffirming the intertwined role played
between both sets of matrix degrees of freedom. We solve the accompanying
Smoluchowski-Fokker-Planck equation valid for any initial matrix. We derive
evolution equations for the averaged extended characteristic polynomial and for
a class of $k$-point eigenvalue correlation functions. From the latter we
obtain a novel formula for the eigenvector correlation function which we
inspect for Ginibre and spiric initial conditions and obtain macro- and
microscopic limiting laws.
