Abstract: We investigate the turnpike problem in optimal control, in the context of time-evolving shapes. We focus here on the heat equation model where the shape acts as a source term, and we search the optimal time-varying shape, minimizing a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We provide necessary conditions for optimality in terms of usual adjoint equations and then, thanks to strict dissipativity properties, we prove that state and adjoint satisfy a measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to an optimal solution of an associated static problem. We illustrate the turnpike phenomenon in shape design with several numerical simulations.