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If farthest galaxies run away from us with acceleration making them exceed speed of light, we should expect them to disappear from sky among time with increasing quantity. Did we observe this? Can we indicate next galaxies to eliminate and their time of decline?
My question concerns galaxies moving with all speed ranges, not only ones greater than speed of light.
No. In fact the opposite is the case.
(See the last paragraph for an intuitive explanation.)
It is a common misbelief that galaxies receding faster than the speed of light are not visible to us. This is not the case; we easily see galaxies moving at superluminal velocities. This does not — as I think most people would think — contradict the theory of relativity, which says that nothing can travel through space faster than $c$. Galaxies do not travel through space (except with small velocities of 100-1000 km/s); rather, space itself is expanding, causing distances between the galaxies to increase.
The recession velocity $v_\mathrm{rec}$ of a galaxy is given by Hubble's Law: $$ v_\mathrm{rec} = H_0 \, d, $$ where $H_0 \simeq 67.8\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ is the Hubble constant (Planck Collaboration et al. 2016). This law implies that galaxies farther away than $$ r_\mathrm{HS} \equiv \frac{c}{H_0} \simeq 4400\,\mathrm{Mpc} \simeq 14.4 \, \mathrm{Gly}\,\,\mathrm{("\!\!Giga\mbox{-}lightyears\!\!")} $$ recede faster than $c$. Here, the subscript "HS" is chosen because the ragion within which galaxies recede slower than $c$ is called the "Hubble sphere". Objects at a distance of $r_\mathrm{HS}$ have a redshift of $z\simeq1.6$.
Consider a photon emitted from a distant galaxy (say, GN-z11 at redshift $z=11.1$) in the past, in the direction of the Milky Way (MW). What special relativity tells us is that locally, the photon always travels through space at $v=c$. Initially, the photon thus increases it distance from GN-z11 at velocity $c$. However, even though the photon travels toward us, its distance to MW increases, due to the expansion of the Universe. As the photon increases its distance to GN-z11, the same expansion causes it to recede from GN-z11 at an ever-increasing velocity. Moreover, as it travels toward MW, it will slowly "overcome" the expansion until it reaches the point where $v_\mathrm{rec} = c$. For an infinitesimally small period, it will stand will wrt. MW, after which it will begin to travel faster and faster as measured from MW. Eventually, its velocity — still in MW's reference frame — will reach $c$, at which point it will have reached MW.
Thus, even though GN-z11 and MW recede from each other at $v_\mathrm{rec} = 2.2c$, we are still able to see it. What is perhaps even more counterintuitive is that when GN-z11 emitted the light we see today, it receded even faster, at $v_\mathrm{rec} \sim 4c$.
There is, however, a limit to how fast a galaxy visible to us can recede, given by the distance $r_\mathrm{PH}$ that light has had the time to travel since the Universe was created. Light comes to us from all directions, so we're situated in the center of a sphere of radius $r_\mathrm{PH}$. This sphere is called "the observable Universe", and its surface (which is not a physical thing) is called the particle horizon (hence the subscript "PH"). Galaxies at the particle horizon are receding at $v_\mathrm{rec}\simeq3.3c$.
As time goes by, light from ever-more-distant galaxies$^\dagger$ will reach us; that is $r_\mathrm{PH}$ increases. In other words, the observable Universe always increases in size, and no galaxy visible today will ever leave the observable Universe, no matter its speed.
However, since future observable galaxies will be more and more redshifted, their light will eventually shift out of the visible range and into longer and longer radiowaves. Furthermore, the time between each detected photon will increase, so they will be dimmer and dimmer, and thus in practice, they will disappear.
A good analogy for better understanding why light can reach us from a galaxy that recedes faster than light, is the "worm on a rubber band": Attach an (infinite stretchable) rubber band (of length, say, 10 cm) to a wall and walk away at any constant speed you choose, e.g. 1 m/s. Before you start, put your pet worm at the end near the wall. It wants to get back to you, and starts crawling at 1 cm/s, i.e. 100× slower than you. Will it ever reach you? If you look at it from the perspective of the wall, both you and the worm move away, but whereas you recede at a constant speed, the worm, although slower in the beginning, accelerates because it moves on the rubber band, but the part of the rubber band between the worm and the wall increases in size. The rest of the rubber band of course also increases in size, but that doesn't matter — as long as you have a constant speed, and the worm accelerates, it will reach you (although in this example, it will take the worm $10^{26}$ billion years, at which point it may have lost its patience. But if you walk at only 10 cm/s, it will take just 6 hours).
In this analogy, you're the MW, the wall is GN-z11, and the worm is a photon. Now if you don't walk at a constant speed, but also accelerate (this is an analogy of the effect of dark energy), the worm may or may not reach you, depending on your speeds. Just like there is a limit to how distant galaxies we will ever be able to see.
$^\dagger$Note that since large distances also means looking back in time (since the light has spent a long time traveling), we actually don't see galaxies this far away, as they hadn't formed this early in history. We do however see the gas from which the galaxies were born, as far back as 380,000 years after Big Bang.
As time passes, there are galaxies that are currently not in the observable universe which will become observable But this is not a sudden winking on. Instead, over hundreds of millions of years we will see a proto galaxy evolve into a mature galaxy.
For example there is a "blob" of hydrogen that some interpret as being the accretion of hydrogen onto a dark matter halo. If this interpretation is correct, then the galaxy that eventually forms from it is outside the observable universe. But it won't remain so. Over billions of years the hydrogen will have formed stars, and the galaxy will be in our observable universe. We don't see the sudden appearance of a new galaxy, rather we see the evolution over billions of years.
There is an effect of greater red-shifting. Ultimately galaxies will begin to retreat fast enough that they are red-shifted below the level of detectability. It is suggested that in about 2 trillion years only local galaxies will be visible. This again is not a rapid process(!)
Thus we do not observe galaxies disappearing over a cosmic horizon, and do not expect to do so.
