The dynamics of two-dimensional (2D) pancake vortices in Josephson-cou
pled 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 s
trongly on the conductivity of normal-metal layers. For a tilted vorte
x line consisting of 2D vortices the equation of viscous motion in the
presence of a transport current parallel to the layers is obtained. T
he specific structure of the vortex line core leads to substantial dev
iations from the Bardeen-Stephen theory. The viscosity coefficient is
found to depend essentially on the angle gamma between the magnetic fi
eld B and the c axis normal to the layers. For field orientations clos
e to the layers the nonlinear effects in the vortex motion appear even
for slowly moving vortex lines (when the in-plane transport current i
s much smaller than the Ginzburg-Landau critical current). In this non
linear regime the viscosity coefficient depends logarithmically on the
vortex velocity V.