One way to answer this is to say that conservation of energy is a law intended to be applied in a single reference frame-- it is not intended to work if you change reference frames. It is true that general relativity makes this an even more difficult point, because it's not just changing reference frames there, but a simpler question appears even in special relativity. If a source of light is moving away from you, and that source is turned on for awhile and then turned off, receivers who are moving away from that source will detect less total light energy than those not moving away. This is true even if the source is a beam that goes entirely into the detector. Thus you could equally ask, if the detector moves away, and detects less energy, where did the missing energy go?
The answer is, there is not any missing energy. The detector moving away always reckoned there was less energy in that beam, and that less energy was conserved the whole time. Conservation of energy is within a given reference frame-- even the energy of a bullet gets much less if you change to the reference frame of the bullet.