We present a continuous-time link-based kinematic wave model (LKWM) for dynamic traf- fic networks based on the scalar conservation law model. Derivation of the LKWM involves the variational principle for the Hamilton-Jacobi equation and junction models defined via the notions of demand and supply. We show that the proposed LKWM can be formulated as a system of differential algebraic equations (DAEs), which captures shock formation and propagation, as well as queue spillback. The DAE system, as we show in this paper, is the continuous-time counterpart of the link transmission model. In addition, we present a solution existence theory for the continuous-time network model and investigate continuous dependence of the solution on the initial data, a property known as well-posedness. We test the DAE system extensively on several small and large networks and demonstrate its numerical efficiency.