This talk focuses on varying boundary optimal control (vbOCPs) governed by partial differential equations (PDEs). This framework features specific parameters changing the portion of the boundary of the spatial domain where the control variable acts to steer the classical solution of the PDE towards a beneficial state. The proposed setting may represent several physical phenomena in many research fields such as energy engineering and geophysics (heat exchangers with baffles, moving fractures...). These disciplines need fast and reliable simulations for many parametric instances to deal with time-consuming tasks such as uncertainty quantification, parameter estimation, and shape design. We propose model order reduction to deal with tasks, building a low-dimensional framework to perform accurate simulations in a shorter amount of time. However, bOCPs usually present transport phenomena and moving fronts: it is well known that classical model order reduction techniques are not appropriate in this context. For this reason, we adapt proper orthogonal decomposition (POD) reduced approaches based on domain transformation or local basis functions, historically used for challenging problems such as hyperbolic ones or wave-like behaviours, to our context. We compare the novel algorithms with standard POD in terms of accuracy and efficiency on two numerical settings with geometries of increasing complexity showing the benefits of using the proposed tailored strategies [1]. [1] M. Strazzullo, F. Vicini, POD-based reduced order methods for optimal control problems governed by parametric partial differential equation with varying boundary control, submitted, 2023, https://arxiv.org/abs/2212.10654.