To get too much more specific, we need to ponder the mass of the black holes and their distance of separation.
You did specify that these black holes were of equal size. They would orbit each other, potentially for billions of years, just like any two other massive objects and how these orbits behaved would depend on their mass, orbital distance, relative velocity and the gravitational influence of any other large bodies. For example, two 30 solar mass black holes orbiting close to Sagittarius A* (our galaxy’s central super massive black hole) would have a very different orbital pattern from the same two black holes orbiting each other in intergalactic space.
We know that black hole mergers are a thing, as LIGO has detected gravity waves from these exact events.
To get too much more specific, we need to ponder the mass of the black holes and their distance of separation.
You did specify that these black holes were of equal size. They would orbit each other, potentially for billions of years, just like any two other massive objects and how these orbits behaved would depend on their mass, orbital distance, relative velocity and the gravitational influence of any other large bodies. For example, two 30 solar mass black holes orbiting close to Sagittarius A* (our galaxy’s central super massive black hole) would have a very different orbital pattern from the same two black holes orbiting each other in intergalactic space.