When a dark matter halo moves through a background of dark matter particles, self-interactions can lead to both deceleration and evaporation of the halo and thus shift its centroid relative to the collisionless stars and galaxies. We study the magnitude and time evolution of this shift for two classes of dark matter self-interactions, viz. frequent self-interactions with low momentum transfer (e.g. due to long-range interactions) and rare self-interactions with high momentum transfer (e.g. contact interactions), and find important differences between the two cases. We find that neither effect can be strong enough to completely separate the dark matter halo from the galaxies, if we impose conservative bounds on the self-interaction cross-section. The majority of both populations remain bound to the same gravitational potential and the peaks of their distributions are therefore always coincident. Consequently any separation is mainly due to particles which are leaving the gravitational potential, so will be largest shortly after the collision but not observable in evolved systems. Nevertheless the fraction of collisions with high momentum transfer is an important characteristic of self-interactions, which can potentially be extracted from observational data and provide an important clue as to the nature of dark matter.