This work proposes an efficient numerical strategy for reproducing crack propagation mechanisms within nano-filled composite structures. The proposed approach consists of a standard Finite Element (FE) set enhanced by a Moving Mesh (MM) technique consistent with the Arbitrary Lagrangian-Eulerian (ALE) formulation. The MM accurately reproduces the evolution of the computational domain geometry due to arbitrarily growing cracks. In particular, the computational mesh is adjusted consistently with the evolution of cracks within the material. The ALE formulation ensures the consistency of motion of mesh nodes, thereby avoiding significant distortions of the finite elements and substantially reducing the need for remeshing actions during the simulation. Mesh nodes move along the crack propagation direction, which is determined by classic fracture criteria typically expressed in terms of Stress Intensity Factors (SIFs) at the crack front. The proposed method adopts the interaction integral method (i.e., M-integral method) to extract fracture variables at the crack front. In particular, the ALE formulation of the M-integral is employed, allowing the use of the M-integral on deforming elements without sacrificing accuracy. The validity of the proposed strategy has been assessed through comparisons with numerical data available in the literature.