The paper presents a nonlinear approach to investigate the behavior of composite sandwich structures with transversely compressible core, under static and dynamic loading conditions. The proposed model, formulated in the 2D framework, incorporates moving mesh cohesive modeling, crack initiation and nucleation at core/skin interfaces. Interface elements are used to predict debonding mechanisms, whereas shear deformable beams and two-dimensional plane stress elements identify skin and core behavior, respectively. In this framework, interfacial crack onset, layer kinematic and debonding propagation effects are correctly simulated. The moving mesh technique, combined with a multilayer formulation, ensures a reduction of the computational costs, required to predict crack onset and progressive evolution of debonding phenomena. Cohesive models for sandwich core/skin interfaces are calibrated by means of comparisons with numerical and experimental data with respect mode I and mode II configurations. Moreover, a parametric study to address the influence of the loading rate and sandwich characteristics on both static and dynamic frameworks is proposed.