This paper investigates a leader-following consensus problem for nonlinear multi-agent systems (MASs) with measurement noises under fixed and Markovian switching topologies, respectively. Noises are considered when each agent measures the states of its neighbours, where intensities of noises are vector functions of relative states. To alleviate the utilization of communication and computation resources, a dynamic event-triggered consensus protocol is designed, where the coupling strength is restricted in a given interval. By using the stochastic stability theorem, it is shown that leader-following consensus is achieved under fixed topology, where an estimation of the convergence rate is given. Moreover, the problem is studied under switching topologies subjecting to the Markovian process, which is applicable to practical situations with a time-varying communication environment. Finally, simulation examples are given to show the correctness of the proposed results.