Angels have the Phone Box
I just had another chat with another MS Answer Tech and he confirmed things:
Do I retain transfer rights of a qualifying Retail license if I upgrade using the free W10 upgrade?
So, I install the qualifying retail license on the new computer. The old computer would be deactivated, (I remove the OS and retire the computer, or install a different OS on it), and I would probably have to call MS to have the new system activated. Once activated, I can upgrade to W10, if the free upgrade offer is still in effect. If it's past July 28, 2016, then I would need to buy a key to upgrade.
Edit: I have captured this chat as well, and if someone wants me to post it here, I will.