The Functionalized Cahn-Hilliard (FCH) energy is a higher-order free energy that has been proposed to describe phase separation in blends of amphiphilic polymers and solvent. It balances interfacial solvation energy of ionic groups and volumetric counter-ion and polymer chain self-interaction energy against elastic energy of the underlying polymer backbone. It is hoped that its gradient flows describe the formation of solvent accessible network structures, such as found in polymer electrolyte membranes, lipid membranes, and amphiphlic diblock copolymers. The FCH gradient flows possess long-lived network morphologies of distinct co-dimension and we characterize their geometric evolution, bifurcation and competition through a formal asymptotic reduction. This reduction encompasses a broad class of coexisting network morphologies with different inner structure, such as bilayers and pores. The stability of the different network morphologies is characterized by the meandering and pearling modes associated to the linearized system. For the H^{-1} gradient flow of the FCH energy, using functional analysis and asymptotic methods, we derive a sharp-interface geometric motion which couples the flow of co-dimension 1 and co-dimension 2 network morphologies, in R^3, through the spatially constant far-field chemical potential. In particular, we rigorously characterize the pearling eigenvalues for a class of admissible co-dimension 1 and co-dimension 2 networks.
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