We have carried out a systematic investigation of the kinetics of domain growth of a model binary mixture in contact with a fractal network similar to the fumed silica network used in recent experiments with polymer blends. This network has selective affinity for one component of the blend. We study the morphology of the growing domains and the wetting layer, the growth exponent, and dynamical scaling behavior for a critical composition of the mixture. We find that the characteristic size of domains grows as t(1/3) for deep quenches, as long as the average domain size is small compared to the average "pore" radius of the unoccupied region. This suggests that the kinetics of domain growth at intermediate times is not appreciably perturbed by the presence of the network. For off-critical compositions, we study domain growth in two different situations where either the majority or the minority component of the blend wets the network. When the majority component wets the network, a network-induced nucleation is possible for sufficiently off-critical mixtures due to the reduction or elimination of the local nucleation barrier for the minority phase near the network. For this reason, minority droplets nucleate predominantly near the network before they appear in the bulk. When the minority phase wets the network, a slowdown in wetting layer growth is found to occur at late times due to a depletion of the system of its minority component.