From what I know, distant galaxies travel with higher speed compare with nearby galaxies.
Distant galaxies also younger compare with nearby galaxies.
Our universe is about 13.75 billion year olds.
When we see galaxies which are 13 billion light year away from us through telescope,
the galaxies we see are actually only 750millions year olds.
and they travel at higher speed compare with (example) a nearby galaxy Andromeda, and Andromeda is about 13.5 billion year olds.
Isn't this show that as galaxies getting older, it's travel speed will also gradually decrease?

I'd think its a matter of the distance we are looking at objects. The further away the object the less accurate the measurements.

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In a rich man's house there is no place to spit but his face.
Diogenes Of Sinope

When astronomer observe that quasars happen only in very distant galaxies, they made a conclusion:
"Because of the great distances to the furthest quasars and the finite velocity of light, we see them and their surrounding space as they existed in the very early universe."

And now the astronomer's observation is that 'high speed galaxies' happen in very distant galaxies.
Can't they make the same conclusion on the 'high speed galaxies'?

When astronomer observe that quasars happen only in very distant galaxies, they made a conclusion:
"Because of the great distances to the furthest quasars and the finite velocity of light, we see them and their surrounding space as they existed in the very early universe."

And now the astronomer's observation is that 'high speed galaxies' happen in very distant galaxies.
Can't they make the same conclusion on the 'high speed galaxies'?

An object in motion tends to stay in motion.

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in a vacuum of course. which space isn't a complete vacuum. after several billion years I'd expect some slowing though gravity from super clusters of galaxies can clearly cause some galaxies to move faster towards the super cluster.

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In a rich man's house there is no place to spit but his face.
Diogenes Of Sinope

Is something supposed to be traveling at a speed exceeding that of light?

Light is supposed to be particles. They could behave as waves as well, if I am not wrong.

When we look at remote galaxies, galaxy clusters and quasars as well, which may be thought of as being active or exploding galaxies, we are supposedly looking back in time.

The appearance of such objects comes from observation made by means of the light received from them which is in the visual range, similar to the colors in the rainbow, or to be more precise, the color spectrum.

We have the blue color in the left part of the color spectrum, assuming wavelengths, everything which is to the left of the blue color relate to ultraviolet, gamma-ray, X-ray and other types of cosmic radiation coming from objects like supernova(e)s, active galaxies (their nucleuses or nuclei), the mentioned quasars and so on.

Everything that lies to right of the red color in the radio spectrum and is having the higher wavelengths (frequencies), relate to the infrared (heat), possible micro-waves, radio waves (where you may find human and possibly other forms or types of communication as well, including possible extraterrestrial communication).

Objects which are known to be very distant, gets dimmer because they are both distant and faint.

What may have started as possible invisible X-ray radiation ends up here as being visible as a reddish glow.

One might assume that the properties of the particles responsible for the radiation may be changing its behaviour or characteristics.

If we observe two quasars lying 13.7 billion light years in opposite directions of the sky relating to each other, would possible inhabitants existing in these two quasars be able to detect either the Milky Way galaxy lying in between and possible each other at the very ends?

Could we possibly assume that the possible inhabitants of the first qusar could not detect the possible inhabitants of the second quasar lying in the opposite end, because either the light is to faint to be detected (meaning dimmed out in the red), or because the combined speed (recession) of each object compared or related to the other object exceeds the speed of light, which is a constant.