We study a nonlinear age-structured model incorporating random diffusion and advection. The model features a monotone nonlinearity affecting the birth rate. We first provide a sufficient and necessary condition for the global dynamics of the model, then explore the asymptotic profiles of the positive equilibrium when it exists in the scenarios of large advection, small or large diffusion, which reveal the effect of diffusion and advection on the spatial distribution of the species. One of our main tools is the principal spectral theory of linear age-structured operators with diffusion and advection. In particular, our results on an associated principal eigenvalue problem, which have independent interest, are crucial in our analysis.