They have a fleshy root system and I have lifted big clumps and taken quite a lot of divisions off them. The smaller you can make the divisions the better chance they have of growing away and flowering sooner. We were working on a lovely old walled garden which had hundreds of varieties of peonies and we moved a lot of them. I had them flowering again the next year. I think a lot depends on the size of the clump and whether you have a big enough piece to send up a flowering bud in it.
Peonies are a fantastic investment.
They'll probably outlive you and they look better and better every year. I do think that when peonies are moved it really helps if they're kept very wet the first and second year after moving. I think that's the biggest problem with them. I'm very dry. But joking aside, in most cases they'll enjoy a moist soil.