Q: When the clients check in, what is the first attempt?
Use the default management server [SEPM server] is the first source for the clients.
AS per the Liveupdate policy setting you configured
1st Priority goes to SEPM, If SEPM is not available then only it move on to Internal Liveupddate server
Q: Does the internal live LiveUpdate server update the clients using delta's?
A: LUA will provide Delta's [Thanks for the update @pete]
Suggestion:
You can implement multiple GUPS on every location and you can increase the content revisions as well on the SEPM. So Delta's will be available for the Clients. It will reduce the Network Bandwidth issue.
If you configure the GUP on the network then the clients contact:
1st source as GUP
If GUP not available then clients move on to
2nd Source as SEPM Server
If SEPM not available then clients move on to
3rd Source Internal/External Liveupdate Administrator
This how Liveupdate designed to work.