We introduce and solve the stabilization problem of a transport partial differential equation (PDE)/nonlinear ordinary differential equation (ODE) cascade, in which the PDE state evolves on a domain whose length depends on the boundary values of the PDE state itself. In particular. we develop a predictor-feedback control design, which compensates such transport PDE dynamics. We prove local asymptotic stability of the closed-loop system in the C(1 )norm of the PDE state employing a Lyapunov-like argument and introducing a backstepping transformation. We also highlight the relation of the PDE-ODE cascade to a nonlinear system with input delay that depends on past input values and present the predictor-feedback control design for this representation as well.