We consider Ising dipolar ferromagnets in a simple geometry (infinite slab of thickness D). When D increases, these systems undergo phase transitions characterized by the appearance of branched domain structures. We have studied the field-induced distortions of the highly branched case, in the framework of the self-similar Privorotskii model. This enables us to calculate the response of the system to a small field step, using a constrained dynamical model of Palmer et al. Relaxations follow a lnt law. This behavior is to be contrasted to the stripe (unbranched) case, where a usual exponential relaxation is obtained. This slowing down can be traced back to the existence of many length scales in the branched regime. Analogous results should hold for dipolar metamagnets.