We first develop the potential and fluctuation theories of continuous-time skip- free Markov processes, extending the recent work from Choi and Patie for skip-free Markov chains. On the one hand, this enables us to revisit in a simple manner the fluctuation theory of continuous-time skip-free random walk on Z. This was originally developed by Spitzer by means of the Wiener-Hopf factorization and, up to now, was the only class of Markov processes with jumps for which such a characterization was attainable. As the second application, we solve the two-sided exit time problems for continuous-time branching processes with immigration (CBI process), which was left open in the literature of this classical family of Markov processes. Next we aim to extend the results to continuous state space branching process with immigration. We identify an intertwining relationship between the discrete and continuous branching pro- cess with immigration. By applying the intertwining relation to the results in discrete CBI, we can derive the first hitting and first passage time of continuous CBI process. Lastly, we briefly introduce the main idea of scaled limit approach, which is an alternative way to study the continuous CBI process.