arXiv:2611.14428v1 Announce Type: cross Abstract: We study explicit Wasserstein--Fisher--Rao descent for fitting Gaussian location mixtures by nonparametric maximum likelihood. The full sequence converges weakly and has finite length in the bounded-Lipschitz metric throughout the original step-size range. These conclusions hold for any fixed data set in any dimension and any initial probability measure. By an earlier conditional optimality theorem, the limit maximizes the likelihood under full-support initialization. Fix any particle step size in $(0,1]$. Initialize the particles by resampling the observations. For an open set of triples of points in the plane, the probability of convergence to a strictly suboptimal fixed point tends to one exponentially as the particle count grows. Without reweighting, a centered Gaussian initialization can give a strictly suboptimal limit. Every finite iterate has a smooth, strictly positive density. The limit assigns positive mass to a nonglobal local maximum of the associated Gaussian sum. We also give the exact pure-transport limit for a one-dimensional initialization with full support and an atom. An exact finite-dimensional representation preserves the truncated likelihood moments and the retained region's contribution to the current dissipation. This lets us transfer an analytic gradient inequality to the measure trajectory.