Tube model predictive control (tube MPC) utilizes an ancillary controller to maintain the state of a uncertain system inside a tube whose central path is generated by an MPC controller. However, for general nonlinear constrained systems, it is difficult to construct such an offline controller. Moreover, since the input space is not fully utilized in central path generation, the closed-loop response could be slow during transitions. In this work, we develop a differential Lyapunov-based tube MPC to overcome these difficulties. It consists of two MPC controllers: (1) a central path generator (CPG) that determines the central path based on the measured states and nonlinear dynamics of the nominal model; (2) an online ancillary controller (OAC) that uses tracking MPC with an additional differential Lyapunov constraint to steer the state of the uncertain system towards the central path. Moreover, an efficient online central path reconstruction strategy based on a path-following algorithm is developed to ensure that the central paths generated by the CPG satisfy the differential Lyapunov constraint to guarantee the feasibility of the OAC. The equilibrium-independent robustness property of the proposed approach is established by using the concepts of contraction and input-to-state stability. An illustrative example is presented, which shows that the proposed approach can provide robust stability as well as fast responses during the transitions. (C) 2018 Elsevier Ltd. All rights reserved.