Learning the dynamics of complex systems features a large number of applications in data science. Graph-based modeling and inference underpins the most prominent family of approaches to learn complex dynamics due to their ability to capture the intrinsic sparsity of direct interactions in such systems. They also provide the user with interpretable graphs that unveil behavioral patterns and changes. To cope with the time-varying nature of interactions, this paper develops an estimation criterion and a solver to learn the parameters of a time-varying vector autoregressive model supported on a network of time series. The notion of local breakpoint is proposed to accommodate changes at individual edges. It contrasts with existing works, which assume that changes at all nodes are aligned in time. Numerical experiments validate the proposed schemes.