1. A mill cannot grind with the water that is past.

2. What is gone is gone.

3. Let bygones be bygones.

4. Let it go and move on.

5. Do not mention the past things; look on the bright side.

6. Forget the past and start anew.

7. Let it be.

Are these all the same meaning?

More or less elinor. there is also "It's no good crying over spilt milk"
Yes. 4, 5, and 6 mention carrying on. The others don't.
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I heard another one "What is the past the past".
 Avangi's reply was promoted to an answer.
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#3, "let bygones be bygones," in my experience is used like "forgive and forget," that is, to forgive a personal grudge or misunderstanding between two people, or possibly groups of people.

The others are unspecific, as far as I can remember.