Allowing slender structures to uplift has been used as an efficient means of controlling their seismic response. Oftentimes rocking structures will exhibit some degree of flexibility and cannot be adequately represented by single-mass or rigid systems. In this paper, we examine the dynamic response of multistory flexible rocking bodies equipped with inerters at their ground level. First, numerical models of inerter-equipped rocking structures are formulated and validated. These models are used to assess the effect of the inerter on the elastic deformations and base rotations demands of a set of structures ranging from 3 to 9 stories. Importantly, we examine the interaction between impact forces and higher vibration modes and evaluate the effectiveness of the inerter in controlling the associated acceleration demands and increased bending moments along the height. The efficiency of the inerter was not found to be affected by practical variations of the stiffness of the rocking surface. Although the inerter increased the moment demands at the first level, the proposed strategy successfully controlled seismic demands along the height of the structure.