  We study synchronization of locally coupled noisy phase oscillators which
move diffusively in a one-dimensional ring. Together with the disordered and
the globally synchronized states, the system also exhibits several wave-like
states which display local order. We use a statistical description valid for a
large number of oscillators to show that for any finite system there is a
critical spatial diffusion above which all wave-like solutions become unstable.
Through Langevin simulations, we show that the transition to global
synchronization is mediated by the relative size of attractor basins associated
to wave-like states. Spatial diffusion disrupts these states and paves the way
for the system to attain global synchronization.
