The problem of leader-following consensus of nonlinear agents in hybrid varying directed topology is considering not only the agent but also that the directed edges can have a time-varying nonlinear dynamics with jump discontinuity, which contains the switching topology as its special case. This paper has the following contributions toward this problem. The leader-following consensus problem in hybrid varying directed topology is first addressed, and an online leader switching method is proposed, which reduces the dependence on some global conditions and the connectivity assumptions on the selection of leaders. Second, we generalize the Lipschitz condition and the combined condition of one-sided Lipschitz and quadratically inner-boundedness conditions to a new generalized linear incremental condition, which gives us a more generalized result in the Lyapunov proof and better performance in simulation. Third, an observer-based consensus protocol is constructed with two sufficient observability and controllability conditions and two optimal control design algorithms. Finally, an example of teleoperating multirobotic manipulator joint network is provided to illustrate the performance improvement by comparing with the existing results.