  The microscopic control available over cold atoms in optical lattices has
opened new opportunities to study the properties of quantum spin models. While
a lot of attention is focussed on experimentally realizing ground or thermal
states via adiabatic loading, it would often be more straightforward to prepare
specific simple product states and to probe the properties of interacting spins
by observing their dynamics. We explore this possibility for spin-1/2 and
spin-1 models that can be realized with bosons in optical lattices, and which
exhibit \textit{XY}-ferromagnetic (or counterflow spin superfluid) phases. We
consider the dynamics of initial spin-rotated states corresponding to a
mean-field version of the phases of interest. Using matrix product state
methods in one dimension, we compute both non-equilibrium dynamics and
ground/thermal states for these systems. We compare and contrast their
behaviour in terms of correlation functions and induced spin currents, which
should be directly observable with current experimental techniques. We find
that although spin correlations decay substantially at large distances and on
long timescales, for induction of spin currents, the rotated states behave
similarly to the ground states on experimentally observable timescales.
