September is the best time to dig and move peonies. Depending on how long they have been in your garden, they may have a very large, deep root system and you probably will not get it all out. That is ok. Take what you need and leave some for the new homeowners. 

When you dig them you may have to break off or cut off parts of the larger root ball. Be sure to remove any decaying parts and your divisions should have 3-5  ‘eyes’ each. 

When you replant them, plant them at the same depth they were growing at. Too deep and they will not bloom. Peonies don’t like to be moved so they may not bloom the first year but should come back to bloom in the second and third year. 

Below is information from the Cornel Extension Service. I am also including a publication from Iowa State Extension that should help you get your peonies off to a new start.

Garden Peony
(Paeonia)
For plant increase, rarely needs division. Divisions may wait up to 3 years before blooming
Division time: September
Divisions should have 3 to 5 well- developed eyes (buds for next year’s growth). Plant peonies with the eyes no deeper than 1 inch below the soil line.

https://www.extension.iastate.edu/news/2009/aug/062801.htm