Sure. With only the two black holes it would take an extremely long time because they would just orbit each other with a little bit of energy loss to gravitational waves, but the black holes will also fly through all the other material in the merging galaxies, which slow them down relative to each other. That gets them closer together. It could still take a very long time, however.

In Newtonian physics, two bodies can orbit each other forever. It's stable. They exert gravitational pull on each other but gravity is frictionless so no energy and momentum is lost and it continues forever.

In general relativity, two orbiting bodies will produce gravitational waves that propagated some energy and momentum away. This is a tiny effect in "normal" situations so it would take eons for enough momentum to be lost for the black holes to coalesce. If the black holes do get close together and hit relativistic speeds then the radiated gravitational energy can become enormous and really drop the black holes into each other fast, as in the detected gravitational wave events. This is the last few wild seconds of a process that may have taken billions of years to develop.

However, a real galaxy is not actually a two body problem. The black holes will be interacting with lots of other stars in the combined galaxy. In these interactions, the smaller bodies (stars) tend to gain momentum and can even be flicked right out of the galaxy. The larger black holes will lose momentum and slow down. These effects are sometimes called gravitational "friction". Over many interactions, the black holes will lose sufficient momentum to fall into each other. By this time they will have either absorbed most of stars or flicked them out into intergalactic space, never to be seen again.

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