Fast and efficient simulation strategies are a basic requirement of technologies such as digital twins. Particularly for roadway infrastructure, recent developments have demonstrated that the arbitrary Lagrangian Eulerian (ALE) formulation can be utilized to improve computational efficiency, when simulating the response of such pavement structures subjected to moving loads. It is also well-established in literature, that model order reduction (MOR) techniques significantly enhance calculation speed. This contribution details the combination of both these tools into a novel model order reduced arbitrary Lagrangian Eulerian (MORALE) formulation. Both hyperelastic and viscoelastic material models are considered in this work. Transient simulations of pavement structures subjected to moving loads are then carried out with the developed framework, and these show a significant enhancement in computational speed and efficiency over conventional simulation techniques. Also in this work, a comprehensive breakdown of the computational costs involved in using the MORALE formulation is provided and analysed. Such an efficient and fast simulation framework is of vital importance in technologies such as digital twins of roadway infrastructure (like pavements), as it enables engineers to quickly run what-if analyses and make informed decisions about the management of the structure under consideration.