  We study the $\omega$-limit set of solutions of a nonlocal ordinary
differential equation, where the nonlocal term is such that the space integral
of the solution is conserved in time. Using the monotone rearrangement theory,
we show that the rearranged equation in one space dimension is the same as the
original equation in higher space dimensions. In many cases, this property
allows us to characterize the $\omega$-limit set for the nonlocal differential
equation. More precisely, we prove that the $\omega$-limit set only contains
one element.
