The rotor of a floating offshore wind turbine experiences intricate aerodynamics due to significant motion in the floating foundation, necessitating a holistic understanding through a synergistic blend of experimental and numerical methodologies. This study investigates rotor loads and the emergence of unsteady phenomena for a floating offshore wind turbine under motion. The approach compares a wind tunnel experimental campaign on a moving scale model with large-eddy simulations. Importantly, both experimental and numerical setups were co-designed simultaneously to match conditions and allow a fair comparison. The experimental setup features a 1:148 scale model of the DTU 10MW reference wind turbine on a six degrees of freedom robotic platform, tested in a wind tunnel. Numerically, the LES code YALES2, employing an actuator line approach undergoing imposed motions, is used. Harmonic motions on one degree of freedom in surge and pitch directions are explored at various frequencies. Thrust force variation aligns with quasi-steady theory for both numerical and experimental results at low frequencies. However, higher frequencies reveal the rise of unsteady phenomena in experiments. Large-eddy simulations, coupled with an actuator line approach, provide additional insights into the near- and mid-wake response to imposed motions. This co-design approach between numerical and experimental tests enhances the comprehension of aerodynamic behaviour in floating offshore wind turbines, offering valuable insights for future designs.