  We investigate the dynamics of the Josephson vortex lattice in layered
high-T$_{c}$ superconductors at high magnetic fields. Starting from coupled
equations for superconducting phases and magnetic field we derive equations for
the relative displacements [phase shifts] between the planar Josephson arrays
in the layers. These equations reveal two families of steady-state solutions:
lattices with constant phase shifts between neighboring layers, starting from
zero for a rectangular configuration to $\pi$ for a triangular configuration,
and double-periodic lattices. We find that the excess Josephson current is
resonantly enhanced when the Josephson frequency matches the frequency of the
plasma mode at the wave vector selected by the lattice structure. The regular
lattices exhibit several kinds of instabilities. We find stability regions of
the moving lattice in the plane lattice structure - Josephson frequency. A
specific lattice structure at given velocity is selected uniquely by boundary
conditions, which are determined by the reflection properties of
electromagnetic waves generated by the moving lattice. With increase of
velocity the moving configuration experiences several qualitative
transformations. At small velocities the regular lattice is stable and the
phase shift between neighboring layers smoothly decreases with increase of
velocity, starting from $\pi$ for a static lattice. At the critical velocity
the lattice becomes unstable. At even higher velocity a regular lattice is
restored again with the phase shift smaller than $\pi/2$. With increase of
velocity, the structure evolves towards a rectangular configuration.
