We analyse the horizon structure of families of spacetimes obtained by evolving initial data sets containing apparent horizons with several connected components. We show that under certain smallness conditions the outermost apparent horizons will also have several connected components. We further show that, again under a smallness condition, the maximal globally hyperbolic development of the many-black-hole initial data constructed in Chrusciel and Delay (2002 Class. Quantum Grav. 19 L71–9), or… Expand