An interaction region with head-on collisions is being developed for the ILC as an alternative to the base line 14 mrad crossing angle design, motivated by simpler beam manipulations upstream of the interaction point (IP) and a more favourable configuration for the detector and physics analysis. The design of the post-collision beam line in this scheme involves however a number of technological challenges, one of which is the strength requirement for the electrostatic separators (ES) placed immediately after the final doublet to extract the spent beam. In this paper, we examine in detail the main mechanism behind this requirement, the multi-beam kink instability, which results from the long-range beam-beam forces at the parasitic crossings after the bunches have been extracted. Our analysis uses realistic bunch distributions, the Guinea-Pig program to treat beam-beam effects at the interaction point and the DIMAD program to track the disrupted beam distributions in the postcollision beam line.