  We study the worst-case mixing time of the global Kawasaki dynamics for the
fixed-magnetization Ising model on the class of graphs of maximum degree
$\Delta$. Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show
that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for
all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and
Perkins, we show that the regime of fast mixing does not extend throughout the
regime of tractability for this model: there is a range of parameters for which
there exist efficient sampling algorithms for the fixed-magnetization Ising
model on max-degree $\Delta$ graphs, but the Kawasaki dynamics can take
exponential time to mix. Our techniques involve showing spectral independence
in the fixed-magnetization Ising model and proving a sharp threshold for the
existence of multiple metastable states in the Ising model with external field
on random regular graphs.
