Transferring 10 GB over the internet?

The easiest way would be to use Back to My Mac in iCloud.

Mac OS X: Using and troubleshooting Back to My Mac with your iCloud account on OS X Lion

Enable file sharing on the mini. If everything is set up right, you should be able to see the sharepoints on the MBA whenever you have a network connection. Some gateways might not pass all the packet types required.

I don't understand your response. You're transferring files over a network.

If you can transfer files over a local network with file sharing, then you can do the same thing over the Internet with BTTM, which is effectively just wide-area Bonjour. You haven't explained why it isn't working for you. It works for others.

I assume that the network connection reliability is your main problem.

I recommend you to use file transfer client which is able to keep the connection and reestablich it in case of failure.

Try CyberDuck FTP/SFTP client. You can get it for free from the homesite http://cyberduck.ch/ or buy it from the App Store.

CyberDuck is quite reliable client. You can set a number of retries along with idle times in case of network failure. I use it regularly as incremental backup of my photos over Inetnet (2-5 GB per session) - no problems.

Cyberduck->Preferences->Connection

For better performance use SCP protocol for file transfer in CyberDuck->Preferences->SFTP

Make sure you enabled remote access to your mini via SSH / SCP.

Hope this will help.

Finder invariably fails with this.

Fails how? What happens?

FTP is not secure. Use SFTP instead.

I do not have Lion Server, but for Lion workstation it is enough to enable "remote login" on System Preferences.

Make sure you can connect to the mini via TCP port 22, I mean make sure the router on the far-end allows connection to the mini on that port (configure port forwarding if nneded).

Regards.