We study energies of Modica–Mortola and Ambrosio-Tortorelli type on generalized hypersurfaces in Euclidian space, which are allowed to vary. We prove several compactness and lower bound estimates in the asymptotic limit, corresponding to the sharp interface limit in the case of Modica-Mortola type energies. This includes results for coupling a phase field to the geometry to the surface, and an approximation of the surface itself by a second phase field. We use the concept of generalized BV functions over currents as introduced by Anzellotti et al. [Annali di Matematica Pura ed Applicata, 170, 1996] to give a suitable formulation in the limit and achieve the necessary compactness properties. This is joint work with Benjamin Lledos (Nîmes), Roberta Marziani (Siena), Matthias Röger (Dortmund).