Why is email being downloaded every time I (dis)connect VPN? I've noticed that every time I connect to or disconnect from a VPN connection, Apple Mail downloads all sorts of old email, email that's already been downloaded locally.  I don't believe it's resulting in any duplicates, but I'd like to know what's really going on.  

For what it's worth, I have 6 email accounts and I'm using OS X 10.9.2.  The attached images are indicative of typical behavior.

After connecting to VPN:



After disconnecting from VPN:
It does that likely because you told it to do and it has lots of work to do before actually disconnecting.

For example with those settings:



Since you have 6 accounts with lots of activity, that can take a while.

To see more details for what it is doing, use the menu bar Windows and Show Activity.



A new window will pop up showing you exactly what is going on.

Sometimes using the Activity window you can see which one of your 6 accounts is doing it.

Your email will open 12 communication channels in network (Internet).
2 for each email account, one for sending and one for receiving, as you can imagine that's a lot of traffic than needs to be shut down orderly.

You can use the Connection Doctor- located in the Windows tab in mail (i do not know why is it called Doctor), but there you can see the communications going on, and maybe the one that is having trouble.

The pic you are showing is typical for a email account that has not been give a chance to finish updating, but once it is done it should stop doing that.