In my working experience, many authors write to change their funding project names after the paper is accepted or during the proofread and formal publication period.
To this phenomenon, what are your comments and suggestions?
If there was a misspelling or other error in the project name, then "yes," it is perfectly fine to change the name to the correct one. If the name you listed was already correct as written, I would not change it. Any database of funded projects would show the original name. Changing it could cause confusion.
It is really a tricky question. It is logical to allow the author to correct a wrong founding source or even to add a missing one. On the other hand it could be a problem if an author deliberately provides misleading information for some reason.
I believe that the real question is whether the wrong or missing information about the funding source has an effect on the review process. For example in specific cases a funding source could provide a clue to the reviewer that the funding institution may influence the results of a study. However, I believe that this is an extremely rare case in earth sciences (it could be the case in medical journals e.g.). In most cases the authors just don’t pay much attention to all these small details before the paper acceptance.
In some journals, in order to face this problem, they request from the authors to clearly state the funding sources in the cover letter or with specific questions at the submission phase to attract the authors’ attention on this matter. Maybe this is a good solution.
In Journal of Mountain Science, the authors are required to include their funding source in the title page (not for peer-review) not in the Main document (for peer-review).
My question is whether the authors arbitrarily delete or add fund names are misbehavior.
Your question of " whether the authors arbitrarily delete or add fund names are misbehavior" seems like a clear-cut question of ethics. No one should "arbitrarily" do anything in a publication for a scientific journal.