In this paper we study time-inhomogeneous versions of one-dimensional Stochastic Differential Equations (SDE) involving the Local Time of the unknown process on curves. After proving existence and uniqueness for these SDE under mild assumptions, we explore their link with Parabolic Differential Equations (PDE) with transmission conditions. We study the regularity of solutions of such PDE and ensure the validity of a Feynman-Kac representation formula. These results are then used to characterize the solutions of these SDE as time-inhomogeneous Markov Feller processes.