The parameters of many physical processes are unknown and have to be inferred from experimental data. The corresponding parameter estimation problem is often solved using iterative methods such as steepest descent methods combined with trust regions. For a few problem classes also continuous analogues of iterative methods are available. In this work, we expand the application of continuous analogues to function spaces and consider PDE (partial differential equation)-constrained optimization problems. We derive a class of continuous analogues, here coupled ODE (ordinary differential equation)-PDE models, and prove their convergence to the optimum under mild assumptions. We establish sufficient bounds for local stability and convergence for the tuning parameter of this class of continuous analogues, the retraction parameter. To evaluate the continuous analogues, we study the parameter estimation for a model of gradient formation in biological tissues. We observe good convergence properties, indicating that the continuous analogues are an interesting alternative to state-of-the-art iterative optimization methods.