We use analytical methods to construct the two-parameter Feller semigroup associated with a Markov process on a line with a moving membrane such that at the points on both sides of the membrane it coincides with the ordinary diffusion processes given there and its behavior after reaching the membrane is described by a kind of nonlocal Feller-Wentzell conjugation condition.
"One-dimensional diffusion processes with moving membrane: partial reflection in combination with jump-like exit of process from membrane." Electron. J. Probab. 25 1 - 21, 2020. https://doi.org/10.1214/20-EJP443