That's physically impossible. You can't observe something that's beyond your "light cone", as any galaxy "moving" (it's not really moving, it's the space that's getting expanding) faster than light would be. What you're referring to is the fact that we can confidently predict that galaxies at the edge of the observable universe, which we currently see moving away from us really fast, but as they were in the distant past due to their light taking billions of years to arrive at us, are currently, if we could actually see them where they are right now (again: we can't), "moving" faster than light away from us.
Once a galaxy has moved beyond or "light cone", it's lost forever: you won't see it again even if you try moving towards it at light speed for all eternity.
The Hubble sphere (the place where recession velocities hit the speed of light) is not the same as the particle horizon (our past lightcone at current cosmological time, the boundary of the observable universe) or the cosmic event horizon (our past lightcone at infinite cosmological time, the boundary of the asymptotically observable universe).
Observations indicate that the expansion of the universe is accelerating, and the Hubble constant is thought to be decreasing. Thus, sources of light outside the Hubble horizon but inside the cosmological event horizon can eventually reach us. A fairly counter-intuitive result is that photons we observe from the first ~5 billion years of the universe come from regions that are, and always have been, receding from us at superluminal speeds.
Once a galaxy has moved beyond or "light cone", it's lost forever: you won't see it again even if you try moving towards it at light speed for all eternity.