This work investigates the issue of describing polydispersivity in an Eulerian framework of a disperse spray with potentially mesh movement. In this perspective, the multi-fluid model and and associated numerical schemes provide robust tools (de Chaisemartin (2009)). However, there is a substantial interest for the development of a method able to account for the size distribution of particles without discretizing the size phase space into sections such as in the classical multi-fluid model. The objective is to write a polydisperse spray model in a formulation close to the usual two-fluid one. Eulerian high order size moment methods are well suited for such problems, but the design of numerical algorithms usually faces two difficulties: accuracy and stability. Whereas a multi-fluid approach with a single section would be too coarse, standard two-fluid models only provide an approximated mean diameter which is also too coarse an assumption in order to capture polydispersivity. This issue has been successfully tackled considering a high order size moment method. The vector of successive size moments of a number density function (NDF) over a given size interval belongs to what is called the moment space. But its complex geometry makes difficult to ensure that a moment vector stays in the moment space during the computation. The achievement of this work consists in designing numerical schemes to address these issues. These methods are displayed in the context of evaporation (Massot et al. (2010)) and advection (Kah et al. (2010)) of a dispersed droplet flow. These results are the first step, for our application of interest, of the simulation of an injected polydisperse spray into a combustion chamber. Indeed, downstream the injector, after primary and secondary atomization, the spray consists in a polydisperse droplet population. An accurate description of the droplet size distribution will improve precision on the fuel fraction evaporated in the gaseous phase. These injection cases involve a mobile mesh to deal with the engine operating conditions. Arbitrary Lagrangian Eulerian (ALE) methods provide numerical schemes adapted for problems involving interactions between fluids and a moving structure. In this paper, we present a new formalism which allows to keep the conservation properties of the numerical scheme for the evolution of the moments in the ALE formalism, in order to preserve the moment space. This new developed tool is validated on academic test cases, showing its feasability in the perspective of its implementation in an unstructured flow solver, IFP-C3D (Bohbot et al. (2009)), which is in progress.