Transplanting a mature peony
The preferred time of year to transplant herbaceous peonies is September as the plant begins to go dormant. The new site must have at last 6 hours of sunlight each day through the growing season.
It's important to realize that transplanted peonies will not bloom well the first spring but they should return to full production by year 3 or 4.
Believe it or not, transplant times and methods are the same across the US. "Transplanting Peonies" clearly describes all the details - https://www.extension.iastate.edu/news/2009/aug/062801.htm.