  The dynamics of 2D pancake vortices in Josephson-coupled
superconducting/normal - metal multilayers is considered within the
time-dependent Ginzburg-Landau theory. For temperatures close to $T_{c}$ a
viscous drag force acting on a moving 2D vortex is shown to depend strongly on
the conductivity of normal metal layers. For a tilted vortex line consisting of
2D vortices the equation of viscous motion in the presence of a transport
current parallel to the layers is obtained. The specific structure of the
vortex line core leads to a new dynamic behavior and to substantial deviations
from the Bardeen-Stephen theory. The viscosity coefficient is found to depend
essentially on the angle $\gamma$ between the magnetic field ${\bf B}$ and the
${\bf c}$ axis normal to the layers. For field orientations close to the layers
the nonlinear effects in the vortex motion appear even for slowly moving vortex
lines (when the in-plane transport current is much smaller than the
Ginzburg-Landau critical current). In this nonlinear regime the viscosity
coefficient depends logarithmically on the vortex velocity $V$.
