  We study an ensemble of random walkers carrying internal noisy phase
oscillators which are synchronized among the walkers by local interactions. Due
to individual mobility, the interaction partners of every walker change
randomly, hereby introducing an additional, independent source of fluctuations,
thus constituting the intrinsic nonequilibrium nature of the temporal dynamics.
We employ this paradigmatic model system to discuss how the emergence of order
is affected by motion of individual entities. In particular, we consider both,
normal diffusive motion and superdiffusion. A non-Hamiltonian field theory
including multiplicative noise terms is derived which describes the
nonequilibrium dynamics at the macroscale. This theory reveals a
defect-mediated transition from incoherence to quasi long-range order for
normal diffusion of oscillators in two dimensions, implying a power-law
dependence of all synchronization properties on system size. In contrast,
superdiffusive transport suppresses the emergence of topological defects,
thereby inducing a continuous synchronization transition to long-range order in
two dimensions. These results are consistent with particle-based simulations.
