AFAIK, the transfer is not permissible via LAN.
AFAIK, the transfers are only local via FireWire or Thunderbolt (and you'll need a match between what the old and the new Mac Mini Server supports), or via a USB-connected local disk.
The way I usually do this is with Disk Utility. Creating an external backup on a USB disk - boot the installer DVD on the old Mac Mini Server, and run Disk Utility from the Utilities menu and create an external copy of the disk - then transfer and connect that disk to the new Mac Mini Server. I use this "backup of the original server" as the source, too; I use the original disk in the Mac Mini as the "backup copy". (This obviously assumes a one-disk configuration; on the newer Mac Mini Server boxes, I usually RAID the internal disks.)
And you should be able to shut down and restart the new Mac Mini Server and get back to that transfer section. At worst, you'll restart the installation from the recovery partition, and get back to that transfer prompt that way.