This paper proposes a generalization of the eXtended finite element method (X‐FEM) to model dynamic fracture and time‐dependent problems from a more general point of view, and gives a proof of the stability of the numerical scheme in the linear case. First, we study the stability conditions of Newmark‐type schemes for problems with evolving discretizations. We prove that the proposed enrichment strategy satisfies these conditions and also ensures energy conservation. Using this approach, as the crack propagates, the enrichment can evolve with no occurrence of instability or uncontrolled energy transfer. Then, we present a technique based on Lagrangian conservation for the estimation of dynamic stress intensity factors for arbitrary 2D cracks. The results presented for several applications are accurate for stationary or moving cracks.