In this document we consider the construction of an adaptation technique for conservation laws with source terms based on continuous mesh deformation without re-meshing. The objective of the report is to analyse all the fundamental aspects of the process. These go from the choice of the appropriate ALE (Arbitrary Lagrangian Eulerian) of the PDE, allowing to preserve at the discrete level the appropriate steady state equilibrium, to the choice of the dicretization method, to the definition of the mesh deformation technique and finally of the coupling scheme-mesh deformation. For each step different alternatives are proposed and evaluated. Several improvements with repeat to the methods existing in literature are validated on properly chosen numerical benchmarks.