  We use 3D numerical simulations to explore the phase diagram of driven flux
line lattices in presence of weak random columnar disorder at finite
temperature and high driving force. We show that the moving Bose glass phase
exists in a large range of temperature, up to its melting into a moving vortex
liquid. It is also remarkably stable upon increasing velocity : the dynamical
transition to the correlated moving glass expected at a critical velocity is
not found at any velocity accessible to our simulations. Furthermore, we show
the existence of an effective static tin roof pinning potential in the
direction transverse to motion, which originates from both the transverse
periodicity of the moving lattice and the localization effect due to correlated
disorder. Using a simple model of a single elastic line in such a periodic
potential, we obtain a good description of the transverse field penetration at
surfaces as a function of thickness in the moving Bose glass phase.
