When two black holes merge, their event horizon looks a bit like a spinning blob (see picture below). The fact that the event horizon of a black hole can move kind of confuses me, because some locations which were previously inside the event horizon are now suddenly not inside the event horizon. So what should appear in these regions? Or in more GR terms: for regions in regular (Minkowski) spacetime what 'appears' in that region is determined by its past lightcone. For the regions that I am confused about it appears to me that the past lightcone point entirely towards the black hole. Is that correct and if so how can these regions exist?
First, remember that the notion of an "event horizon" is global statement about the causal structure of spacetime, rather than some local statement. As such talking about the movement of such an object is fraught with difficulty, and any statement about such movement is bound to depend on exactly how one chooses to "slice" spacetime into surfaces of equal "time".
By definition, the event horizon marks the boundary of events in spacetime that can appear in the causal past (past lightcone) of any outside observer. This means that if on any time slice a point lies outside the event horizon, all of the causal past of that point must lie outside of the event horizon.
Now for your intuition. You seem to be led astray by applying Minkowski space intuition to the highly dynamic and strongly curved spacetime around a black hole (merger).
Although I do not necessarily like this language, the simplest way to think about the region of space just "behind" the motion of a black hole is as if this piece of space is being "dragged" along with the black hole. Imagine a ray of light just behind the black hole on a trajectory where it just barely escapes the event horizon. In "reasonable" global coordinates it may appear on a next time slice that this light ray has moved along with the black hole, and now exists at spacial coordinates previously inside the event horizon.