The Arbitrary Lagrangian Eulerian (ALE) formulation in the context of solid dynamics was originally conceived to address the development of excessive mesh distortion in highly deformable hypoelastic-plastic and hyperelastic-plastic processes. In addition to the material and spatial configurations, an additional referential (intrinsic) configuration was introduced in order to disassociate material particles from mesh positions. Due to their ability to circumvent the shortcomings of purely Lagrangian/Eulerian approaches, the use of ALE based methods is still of paramount importance in the context of multi-materials modelling. This paper presents a new computational framework using a novel ALE formalism in the form of a system of first order conservation laws. Using isothermal hyperelasticity as a starting point, mass and linear conservation laws are written and solved with respect to the reference configuration. In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the deformation gradient tensor (from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate into (Total Lagrangian and Eulerian) alternative mixed systems of conservation laws. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. Spatial discretisation will be carried out exploiting two in-house frameworks, namely, vertex centred Finite Volume (i.e. OPENFOAM) [1] and Smooth Particle Hydrodynamics [2]. Second and third order Runge-Kutta time integration schemes will be used to advance the fields in time. An extensive range of three dimensional numerical examples will be presented in order to demonstrate the robustness and reliability of the framework.