We study the asymptotic behavior of solutions of the Cauchy problem associated to a quantitative genetics model with a sexual mode of reproduction. It combines trait-dependent mortality and a nonlinear integral reproduction operator, the infinitesimal model. A parameter describes the standard deviation between the offspring and the mean parental traits. We show that under mild assumptions upon the mortality rate m, when the deviations are small, the solutions stay close to a Gaussian profile with small variance, uniformly in time. Moreover, we characterize accurately the dynamics of the mean trait in the population. Our study extends previous results on the existence and uniqueness of stationary solutions for the model. It relies on perturbative analysis techniques with a sharp description of the correction from the Gaussian profile.