It has previously been observed that the evolution of entanglement entropies and OTOCs in systems ranging from random unitary circuits to holographic CFTs can be understood within a common framework known as the “membrane picture.” In this talk, we consider a class of Brownian local evolutions, where the Lorentzian evolution of the system on multiple copies can be equivalently described by a Euclidean evolution on multiple copies. We identify generic properties of the low-energy modes of the superhamiltonian which lead to the emergence of the membrane picture. We further discuss the breakdown of the membrane picture in integrable systems and systems with conserved quantities due to differences in the low-energy spectrum of the superhamiltonian.