  The dynamics of membrane undulations inside a viscous solvent is governed by
distinctive, anomalous, power laws. Inside a viscoelastic continuous medium
these universal behaviors are modified by the specific bulk viscoelastic
spectrum. Yet, in structured fluids the continuum limit is reached only beyond
a characteristic correlation length. We study the crossover to this asymptotic
bulk dynamics. The analysis relies on a recent generalization of the
hydrodynamic interaction in structured fluids, which shows a slow spatial decay
of the interaction toward the bulk limit. For membranes which are weakly
coupled to the structured medium we find a wide crossover regime characterized
by different, universal, dynamic power laws. We discuss various systems for
which this behavior is relevant, and delineate the time regime over which it
may be observed.
