This manuscript proposes novel distributed algorithms for solving the dynamic consensus problem in discrete-time multi-agent systems on three different objective functions: the average, the maximum, and the median. In this problem, each agent has access to an external time-varying scalar signal and aims to estimate and track a function of all the signals by exploiting only local communications with other agents. By recasting the problem as an online distributed optimization problem, the proposed algorithms are derived based on the distributed implementation of the alternating direction method of multipliers (ADMM) and are thus amenable to a unified analysis technique. A major contribution is that of proving linear convergence of these ADMM-based algorithms for the specific dynamic consensus problems of interest, for which current results could only guarantee sub-linear convergence. In particular, the tracking error is shown to converge within a bound, whereas the steady-state error is zero. Numerical simulations corroborate the theoretical findings, empirically show the robustness of the proposed algorithms to re-initialization errors, and compare their performance with that of state-of-the-art algorithms.