How do I sync added POP3 mail accounts to IMAP client app so I can manage on desktop?

How do I sync added POP3 mail accounts to IMAP client app so I can manage on desktop?

I've added three POP3 accounts to my zoho account.  I can view them all from the zoho Web interface at mail.zoho.com.
 
However, when I set up Thunderbird to fetch mail from my zoho account, only the zoho account's mail synchronized.  How to I make the rest of the mail associated with the zoho account synchronize, so that I can manage all of it?  The web mail interface is okay, but I was trying to have all of my mail manageable from Thunderbird.
 
The mail service where I'm picking up my mail via pop does not appear to offer mail forwarding.  Otherwise, I could forward it from there to my zoho account, and then set up filters in the zoho account to put it into certain folders, or label it, etc.  Then it would be under the umbrella of the zoho account, and thus managed by imap.
 
How do I force zoho mail account to take ownership of the pop3 email it is importing?  Is there a way to move it to a different folder upon receipt?
 
thank you
Linda
 
___________________________ EDITED 6/29/11 5:30 PM ___________________________
 
I learned how to forward mail from my source accounts, so I am attempting to use that instead of POP3.  Now I have the problem that filters do not work on the forwarded as I expected.  Topic: https://help.zoho.com/portal/en/community/topic/only-one-filter-applies 
 
If there is a solution to the original POP3 question above - making the zoho account "own" the mail it gathers via POP3 so it can be managed via IMAP - please let me know.
 
thank you
Linda