We study probabilistic protocols for concurrent threshold-based load balancing in networks. There are n resources or machines represented by nodes in an undirected graph and m >> n users that try to find an acceptable resource by moving along the edges of the graph. Users accept a resource if the load is below a threshold. Such thresholds have an intuitive meaning, e.g., as deadlines in a machine scheduling scenario, and they allow the design of protocols under strong locality constraints. When migration is partly controlled by resources and partly by users, our protocols obtain rapid convergence to a balanced state, in which all users are satisfied. We show that convergence is achieved in a number of rounds that is only logarithmic in m and polynomial in structural properties of the graph. Even when migration is fully controlled by users, we obtain similar results for convergence to approximately balanced states. If we slightly adjust the migration probabilities in our protocol, we can also obtain fast convergence to balanced states.