  In this paper, we consider the long-term behavior of some special solutions
to the Wave Kinetic Equation (WKE). This equation provides a mesoscopic
description of wave systems interacting nonlinearly via the cubic NLS equation.
Escobedo and Vel\'azquez showed that, starting with initial data given by
countably many Dirac masses, solutions remain a linear combination of countably
many Dirac masses at all times. Moreover, there is convergence to a single
Dirac mass at long times. The first goal of this paper is to give quantitative
rates for the speed of said convergence. In order to study the optimality of
the bounds we obtain, we introduce and analyze a toy model accounting only for
the leading order quadratic interactions.
