The Airy$_β$ line ensemble is an infinite sequence of random curves. It is a natural extension of the Tracy-Widom$_β$ distributions, and is expected to be the universal edge scaling limit of a range of models in random matrix theory and statistical mechanics. In this work, we provide a framework of proving convergence to the Airy$_β$ line ensemble, via a characterization through the pole evolution of meromorphic functions satisfying certain stochastic differential equations. Our framework is then applied to prove the universality of the Airy$_β$ line ensemble as the edge limit of various continuous time processes, including Dyson Brownian motions with general $β$ and potentials, Laguerre processes and Jacobi processes.