Hi,
Depending on the way emails are stored in the user's mailbox, you may have to parse the user's inbox (often seen as a regular file were all mails are written one after the other) using sed or something else to process the inbox one mail after the other.
Then you can re-send them using "mail" or "email".
You can also imagine creating a script using pop3 to retrieve the mail on the mailserver if you don't have local access to the mailbox...
As a matter of fact there are MANY solutions. It really depends on what you want to achieve (just moving all the content of your inbox once, or purging only old mails on a regular basis for all your users...).
As an exemple, if you just want to get a copy of your own mails on a second mail server once, then you'd probably be quicker by copying your inbox (~/Mailbox or other, it depends on the soft you use to read your mails) to the second server. As an exemple:
francois@mailserver1 $ scp ~/Mailbox
francois@mailserver2:~/Mailbox.old
francois@mailserver1 $ ssh francois@mailserver2
francois@mailserver2 $ cat Mailbox Mailbox.old > Mailbox.new
francois@mailserver2 $ mv Mailbox.new Mailbox
Not really the nicest, but that's easy to do (provided that I'm using the same mail user agent on the 2 machines cause they musty be stored in the same format, and a backup might be a good idea as well).
Conclusion, if you want a real answer to your question, you should
1/ explain you want to do
2/ explain what soft you use, what permissions you have
...