  We study the asymptotic convergence of solutions of $\partial_t u=-f(u)+\int
f(u)$, a nonlocal differential equation that is formally a gradient flow in a
constant-mass subspace of $L^2$ arising from simplified models of phase
transitions. In case the solution takes finitely many values, we provide a new
proof of stabilization that uses a {\L}ojasiewicz-type gradient inequality near
a degenerate curve of equilibria. Solutions with infinitely many values in
general need not converge to equilibrium, however, which we demonstrate by
providing counterexamples for piecewise linear and cubic functions $f$.
Curiously, the exponential rate of convergence in the finite-value case can
jump from order $O(1)$ to arbitrarily small values upon perturbation of
parameters.
