The initiation, growth, and stability of buckling driven debonding in structural assemblies of stiff blocks, compliant joints, and adhesively bonded composite layers are analytically investigated. The model is developed with focus on masonry walls externally strengthened with composite materials where static and, mainly, dynamic loads may induce compression in the strengthening layers triggering a buckling driven debonding near the joints. The model introduces the interfacial nonlinearity (debonding) through a cohesive interface approach. The geometrical nonlinearity is introduced through the kinematics of intermediate class of deformation (large deflections, moderate rotations, small strains), and the material nonlinearity of the masonry construction is introduced through the constitutive law for the mortar joints. A numerical study of the debonding process in strengthened masonry walls is presented. The study uses the periodicity of the wall for sub-structuring and examines configurations that include composite strips or sheets, strengthening on one face of the wall or on both faces, and compliant mortar materials.