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In this article, the influence of important parameters on the size-dependent thermoelastic behavior of functionally graded magneto-electro-thermo-elastic microcylinders (FGMEE) is studied by means of modified stress couple theory and under the influence of combined mechanical-thermal-magnetic loads. The equations of motion were derived by considering the linear behavior of the shells and considering the first-order theory of the nonlocal shear deformation of the shells. In the following, the results of solving the equations governing the buckle are analyzed. Based on this, firstly, the mechanical properties used in the shell are presented, then the linear buckling behavior is studied, and finally, the size-dependent buckling behavior of these structures is studied. In the presented results, the mode numbers of each buckling load are shown as n and m, where n and m represent the circumferential and longitudinal mode numbers, respectively. In the theory section, using the pairwise size-dependent theory, the stress is corrected, and considering the shell with a relatively thick geometric structure and the forces due to the heterogeneous thermal boundary conditions, a combination of convection heat transfer and displacement is used to obtain a three-dimensional shell temperature distribution. The equilibrium equations of the term electromagnetic coupling system are obtained using the first-order shear displacement field of the shell and the structural relationships in the functionally graded magneto-electro-thermo-elastic (FG-METE) intelligent material. Then, using appropriate analytical and numerical methods, various components of the displacement field, electrical and mechanical potential, and, most importantly, structural stresses for different boundary conditions are obtained.