The dynamics near the symmetry-breaking transition in the sponge phase (L3) of self-assembling surfactant solutions is considered. The surfactant motion is taken to be diffusive (conserved), while the order parameter for the transition is assumed to follow two channels of relaxation: diffusion (conserved) and leakage (nonconserved). Our dynamical treatment is based on mean-field theory within a time-dependent Landau-Ginzburg approach, whose static limit reproduces an earlier successful theory of the static structure factor. We consider two main regimes: in the first, relaxes rapidly compared to the surfactant diffusion, while in the second, the opposite limit applies. We find that in the fast- regime the surfactant dynamical structure factor S(k,t) is exponential in time, but the relaxation rate shows an unusual logarithmic behavior. On the other hand, in the slow-regime, S(k,t) is very nonexponential in time (although the average relaxation rates show the conventional critical slowing-down effects). We argue that a crossover from the fast- to the slow- case occurs as the k vector is increased. Implications for dynamic light scattering from sponge systems are discussed.