The paper aims to analyse the dynamic behaviour of frames fitted with tuned inerter dampers (TIDs), accounting for various aspects of inelastic frame behaviour, bending-torsion (B-T) coupling, irregularities in elevation, and transient nature of the ground motions. An inerter is a mechanical two-terminal device that creates a force proportional to the relative acceleration between its terminals. It has the potential to produce high dynamic mass while its static mass remains fixed. To gain understanding about its working, an inerter was fabricated and installed in a 3-storey shear-frame model for experimentation and numerical validation. Then, we focus on the behaviour of a 5-storeyed inelastic building frame model fitted with a TID on the ground floor and subject to transient earthquake support motion. The presence of TID is observed to be beneficial in terms of reducing response features such as the ductility demand, hysteretic energy absorbed, and measures of global damage. Finally, a one-story frame, asymmetric in plan, was subjected to two-component earthquake support motion to investigate the influence of the location and number of TIDs on the B-T coupled inelastic behaviour of the frame. The study established the beneficial aspects of incorporating TID(s), while also highlighting the need for further parametric study to design and determine the optimal position of TIDs for achieving best performance.