The Slodkowski theorem shows that a holomorphic motion of a subset in the Riemann sphere over the unit disk is extendable. Several proofs of the Slodkowski theorem are given by applying several complex variables. The idea from Chirka uses generalized Beltrami equations, which returns the study to the original idea of Bers and Royden. We have combined both ideas to give a complete theory for the extension problem of holomorphic motions of subsets in the Riemann sphere over hyperbolic Riemann surfaces. In this talk, I will provide an overview of this work. I will discuss the extension problem's zero-winding number and trivial monodromy conditions. They are both necessary. I will give a counterexample to show that the zero-winding number condition is not sufficient and prove that the trivial monodromy condition is sufficient. I will explain how the study of the lifting problem, which used to be called one of the most important problems in Teichmueller theory by Bers and Royden, plays an important role in studying the extension problem.