This paper addresses the distributed consensus tracking problem for a class of switched nonlinear multi-agent systems in pure-feedback form under directed communication networks. For a practical environment, switched non-affine nonlinearities for followers are unknown and their switching signals are assumed to be arbitrary and asynchronous. Compared with existing results in the literature, the main contribution of this paper is to provide a universal control strategy to deal with asynchronously switched nonlinear multi-agent systems in the consensus tracking field. By using the function approximation technique, an approximation-based novel recursive design approach for asynchronously switched followers is presented to design local common adaptive controllers without requiring the knowledge on the signs of control gain functions and on the leader's velocity. From the common Lyapunov function method, it is shown that the consensus tracking errors are asymptotically bounded and converge to a neighborhood of the origin. A simulation example is included to demonstrate the effectiveness of the proposed methodology. (C) 2017 Elsevier Ltd. All rights reserved.