In the general sense, this is like syncing with multiple, possibly unrelated, mail servers -- which is, in the general sense, is meaningless.
Which settings are different? Different mail server addresses? Logins?
Sounds so ugly. And I've never heard about, or used, anything like this before.
If it's the same mail server / account / same messages on the server -- how come the server isn't available for connecting from everywhere, or maybe from the office network and via a VPN when you're at home?
Just curious about the rationale and actual configuration.