Did you read the example that I posted,
here. It walks you through the scenario that you're describing. It works it out in detail.
My comment:
If A is in motion relative to B; then the movement of A relative to B is symmetric to the movement of B relative to A. The answer to your question is that from A's point of view, any clock being carried by B, slows down. And, from B's point of view, any clock carried by A, slows down the same amount.
And your comment on my comment:
Then there is no change in any clock, since relative to any other object, they all are moving the same speed, if there is not a third point of reference.
Your comment is essentially ignoring most of what I said. I don't know if you're being serious, or just kidding.
The answer to your question is that from A's point of view, any clock being carried by B, slows down.That explicitly states that, from A's perspective:
any clock being carried by B, slows down. So, from A's perspective, clocks in B's reference frame are running slower that clocks in A's reference frame. Sure, by symmetry, B has the symmetric experience, namely, it sees clocks in A's time reference as running slow, slower than clocks in its own reference frame. So, yes, there are changes in the clocks; it's just that the changes depend upon your reference.
Because the moon rocket clocks would have slowed down when viewed from earth, and the earth clocks would have slowed down when viewed from the rocket. And if both the faster moving object slows down the same as the stationary, then there would be no time dilation recorded on the moon rocket clocks.
Do you know when they verified the difference in the clocks? While the rocket was in flight? Or, a difference in time after the rocket returned to earth? If, the earth station asked the astronauts what their clock showed and compared it to the earth clock while they were in flight, then they would see time dilation - this explicit case is diagrammed on the reference that I gave you in post 17. If they compared the clocks after the rocket had returned to earth, then, you have to take gravitational time dilation into account (I noted in post 14 that their are 2 components to time dilation). If you want to read the
NASA paper on time dilation on the flight to the moon, you'll notice that they are taking gravitational effects into account.
So if that post is correct, then the shows on time dilation are incorrect, and they are shows used to teach the concept in schools, since I saw it there, and saw a similar show on a physics show on TV.
The website that I referred you to in post 17 is a page that belongs to a physics professor at the University of Virginia,
Michael Fowler. That post is based on one of his lectures. That post supports what I am saying. So, your claim that it disagrees with what is taught in "schools" is not correct.
If you want to know how relative motion effects clocks relativistically, read the reference from post #17 (same reference given at the beginning of this post). Based on your post #18, you never read that reference. If you disagree with something in that example in the reference, please state what it is that you disagree with.