Freezing and melting behaviors of a fluid confined to pores of mesoporous silicon with a modulated structure have been studied using NMR techniques. The molecular self-diffusivities, measured along the freezing and melting transitions, unveiled essential differences in the configuration of the frozen domains. This suggests that freezing is dominated by a pore-blocking mechanism. Freezing kinetics is found to exhibit very slow long-time dynamics, following a ln(2)(t) dependence. This type of time dependence may result if the front of the frozen phase is assumed to propagate in the random potential field created by the disorder of the porous silicon channels, similar to the mechanism of Sinai diffusion. The free energy barriers calculated from the kinetic measurements and estimated using a thermodynamical model yield a consistent picture of the freezing process in the presence of quenched disorder.