Classical model order reduction (MOR) techniques struggle in compressing solution manifolds when local structures travel along the domain. We study MOR algorithms for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description, obtaining improved time saving in the online phase. We work in an arbitrary Lagrangian–Eulerian (ALE) framework in both offline and online phases. We calibrate the solutions aligning the advected features in a reference domain. Then, a classical MOR algorithm (PODEI–Greedy) is used in the ALE framework, while the calibration map is learned through regression techniques. We test the algorithm on scalar one-dimensional hyperbolic problems with various boundary conditions, showing that we outperform classical methods. Finally, we compare the results obtained with different calibration maps.