I am looking into using an LTSP client. I am trying to understand the architecture of such system. Does the main server that it connects to need to be on the local network? For example, would it be possible to plug in an LTSP client into any network. Then have the LTSP client connect to a remote Edubuntu server that is not on the local network?
It depends on the client. If it is a typical thin client, it will need to receive information from the network early in the boot sequence. From the LTSP wiki:
The DHCP server should be local. It is possible to route DHCP traffic, but you would need access to configure every router between the client and the server. It is possible to run the DHCP service on a server other than the LTSP server. I have experience running it on a Windows server.
You will also need to take into consideration the amount of bandwidth needed.