We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and extend well-known variational principles to the weakly convex setting. By resorting to the new enlarged subdifferential, we provide an algorithmic pattern of descent for weakly convex minimization. Under minimal assumptions, we show subsequential convergence to a critical point, and links with difference-of-convex algorithms and criticality conditions are also discussed.