In this paper, nonlinear buckling responses of functionally graded (FG) thin-walled open section beams based on Euler–Bernoulli–Vlasov theory is presented. The finite element incremental equilibrium equations are developed by updated Lagrangian formulation using the non-linear displacement cross-section field that accounts for large rotation effects. Young’s modulus of FG beams are varied continuously through the wall thickness based on the power-law distribution. Numerical results are obtained for thin-walled FG beams with symmetric and mono-symmetric I-section and channel-section for various configurations such as boundary conditions, geometry, skin-core-skin ratios and power-law index to investigate the flexural–torsional and lateral buckling loads and post-buckling responses. The accuracy and reliability of proposed model are proved by comparison with previous research and analytical solutions. The importance of above-mentioned effects on buckling results is demonstrated on benchmark examples.