We provide a general model for Brownian motions on metric graphs with interactions. In a general setting, for (sticky) Brownian propagations on edges, our model provides a characterization of lifetimes and holding times on vertices in terms of (jumping) Brownian accumulation of energy associated with that vertices. Propagation and accumulation are given by drifted Brownian motions subjected to non-local (also dynamic) boundary conditions. As the continuous (sticky) process approaches a vertex, then the right-continuous process has a restart (resetting), it jumps randomly away from the zero-level of energy. According with this new energy, the continuous process can start (or not) as a new process in a randomly chosen edge. The model well extends to a higher order of interactions, here we provide a simple case and focus on the analysis of earthquakes.