  Nonlocally coupled oscillators with a phase lag self-organize into various
patterns such as global synchronization, the twisted state, and the chimera
state. In this paper, we consider nonlocally coupled oscillators that move on a
ring by randomly exchanging their positions with the neighbors, and investigate
the combined effects of phase lag and mobility on the collective phase
dynamics. Spanning the whole range of phase lag and mobility, we show that
mobility promotes synchronization for an attractive coupling, while it destroys
coherence for a repulsive coupling. The transition behaviors are discussed in
terms of the timescales of synchronization and diffusion of the oscillators. We
also find a novel spatiotemporal pattern at the border between coherent and
incoherent states.
