Summarize this article:

This is when the plant will be dormant, making it safer to move it to another spot. Avoid transplanting peonies in the spring when they’re beginning to bloom. August through the beginning of November is a safe period to transplant peonies. In the fall, the peony’s leaves will turn brown and wither. Use a pair of garden shears to cut the stems off near the base of the top of the root ball. You don’t need to keep any stems—as long as the root ball has buds or eyes on it, it will regrow. If you’re transplanting your peonies in the spring, don’t cut the foliage off. If you dig too close to the plant, you could cut into important roots and damage the plant. Try digging a circle around the plant that’s at least 6 in (15 cm) away from the base to make sure the roots are safe. Continue digging a circle around the peonies, loosening up the soil carefully. Loosen the soil until you can grasp the root ball gently and easily lift it from the ground. If you have to pull on the root ball to get it out of the soil, the roots and soil aren’t loose enough yet for it to be safely removed. Shake the plant gently to help extra soil fall off easily. You can also use a hose or watering can to spray the plant, washing the soil off of the roots. Rinsing the plant will help you see the roots and buds so you can replant it correctly. Separate the plant into sections so that each section will have 3-5 buds, or eyes, as well as a healthy portion of root system. Use the knife to cut the plant into as many sections as is necessary.  The buds will look like small white or pink eyes on the root ball. If your peony plant only has 3-5 buds, it doesn’t need to be divided. Use a clean knife to prevent the plant from getting infected.
Aim to transplant the peonies in the fall. Trim away foliage to prepare the plant for new blooms. Dig 6–12 in (15–30 cm) from the peonies to avoid damaging the roots. Use a shovel or spade to loosen the root ball from the ground. Remove excess soil from the roots gently. Use a sharp knife to cut the plant into sections with 3-5 buds each.