  We study synchronization dynamics in binary mixtures of locally coupled
Kuramoto oscillators which perform Brownian motion in a two-dimensional box. We
introduce two models, where in model $\cal I$ there are two type of
oscillators, say $\cal A$ and $\cal B$, and any two similar oscillators tend to
synchronize their phases, while any two dissimilar ones tend to be out of
phase. In model $\cal J$, in contrast, the oscillators in subpopulation $\cal
A$ behave as in model $\cal I$, while the oscillators in subpopulation $\cal B$
tend to be out of phase with all the others. In the real space all the
oscillators in both models interact via a soft-core repulsive potential. Both
subpopulations of model $\cal I$ and subpopulation $\cal A$ of model $\cal J$,
by their own, exhibit a phase coherent attractor in a certain region of model
parameters. The approach to the attractor, after an initial transient regime,
is exponential with some characteristic synchronization time scale $\tau$.
Numerical analysis reveals that the attractors of the two subpopulations
survive within model $\cal I$, regardless of the composition of the mixture
$\phi$ and the strength of the cross-population negative coupling constant $H$,
and that $\tau$ sensitively depends on $\phi$, $H$ and the packing fraction. In
particular, the ability of the oscillators to move and exchange neighbours can
significantly decrease $\tau$. In contrast, model $\cal J$ predicts suppression
of the synchronized state in subpopulation $\cal A$ and emergence of the
coherent attractor in the "contrarians" subpopulation $\cal B$ for strong and
weak cross-population coupling, respectively.
