I asked the same question and got a response that there's no easy way to move the whole site somewhere else.
"We can in theory move some of the content across between bedsparishes and e-voice, for example, but even then any links with the bedsparishes domain in it wouldn't work. This would be mitigated somewhat if you used the same url stub for an e-voice site. (i.e., so you had e-voice.org.uk/THING which matches bedsparishes.gov.uk/THING). But any links referencing bedsparishes directly would need manually updating."
Some items would get moved over but a lot would be lost.
Not sure it helps.
James (a user not an admin)