This paper considers a general setup for the constrained convex optimisation problem over jointly-fully connected time-varying networks. We propose a novel cutting plane-based method that embeds a proximity-based consensus scheme for solving this (potentially non-smooth) optimisation problem. The consensus mechanism allows agents to select the same sample point and therefore reconstruct the centralised cut individually. Under convexity, we prove that agents' sample points converge to the optimal set of the global problem. Numerical tests show the performance of the algorithm.