That's right. This situation however, is a thought experiment. In reality, no energy can just disappear.
The carrier of the electromagnetic field is the photon. It is light, so obviously it travels at the speed of light. In a vacuum, it's always c. In the presence of matter it's always something less.
The carrier of the gravitational field is the graviton. The only way to see a change in the gravitational field is by observing the field of at least 2 oscillating objects. The field thus emanating from the objects will then contain waves, which travel at the speed of light. Since gravity is so weak, these haven't been found yet. Some results should be in sometime next summer, I think.
The sum of the energy of the objects that exist in the universe and the energy of the gravitational field between them is zero.