The paper is devoted to the approximate consensus problem for networks of nonlinear agents with switching topology, noisy and delayed measurements. In contrast to the existing stochastic approximation-based control algorithms (protocols), a local voting protocol with nonvanishing step size is proposed. Nonvanishing (e.g., constant) step size protocols give the opportunity to achieve better convergence rate (by choosing proper step sizes) in coping with time-varying loads and agent states. The price to pay is replacement of the mean square convergence with an approximate one. To analyze dynamics of the closed loop system, the so-called method of averaged models is used. It allows to reduce analysis complexity of the closed loop system. In this paper the upper bounds for mean square distance between the initial system and its approximate averaged model are proposed. The proposed upper bounds are used to obtain conditions for approximate consensus achievement. The method is applied to the load balancing problem in stochastic dynamic networks with incomplete information about the current states of agents and with changing set of communication links. The load balancing problem is formulated as consensus problem in noisy model with switched topology. The conditions to achieve the optimal level of load balancing (in the sense that if no new task arrives, all agents will finish at the same time) are obtained. The performance of the system is evaluated analytically and by simulation. It is shown that the performance of the adaptive multi-agent strategy with the redistribution of tasks among "connected" neighbors is significantly better than the performance without redistribution. The obtained results are important for control of production networks, multiprocessor, sensor or multicomputer networks, etc.