We present a comprehensive framework for the development of gradient flows of parameterization independent surface energies naturally expressed in terms of intrinsic quantities (curvature and metric). To this mix we add cartesian distance which allows adhesion-repulsion energies that guide folding flows of cellular organelles, and surface diffusion of embedded agents (eg proteins). Via a penalty method on membrane density, we derive a mechanism to generate locally incompressible flows of  “fluidic” membranes. In space dimension two, we show that stability analysis of surface patterns can be converted to an analysis of the second variation of the surface energy subject to the nonlinear constraints imposed on the first and second fundamental forms (Gauss Equation and Codazzi-Mainardi equations. We make application to adhesion-repulsion energies that guide folding flows of cellular organelles.