A combined electronic-nuclear wave packet motion is accompanied by temporal changes of probability flux densities. Using a two dimensional model, we study such densities in the vicinity of a conical intersection (CI) between the potential energy surfaces of two electronically excited states. When the dynamics is accompanied by an efficient population transfer, the electronic flux density behaves nearly time-independent although the nuclear flux does not. The second case involves a nuclear motion where the CI is surrounded. There, the nuclear wave packet undergoes a bifurcation, and the electronic density shows characteristics of a rotation. The electronic flux, however, exhibits a constant directional dynamics during the nuclear motion. The geometrical phase which appears in comparing the nuclear dynamics derived from the coupled motion and the Born-Oppenheimer calculation is also seen in the nuclear flux dynamics.