In early to middle September, identify the healthiest red raspberry plant from which to draw suckers. Look around the base of the mother plant for suckers, which are also known as daughter plants. Put garden gloves on, because you will be working around the red raspberry's thorny canes. Use garden shears to snip roots that connect the daughter plant to the mother plant. Be careful to preserve a soil ball around the daughter plant's root system.
Select a full-sun location in your yard in which to transplant your red raspberry plant. Destroy any perennial weeds growing near your raspberry bed and be sure to kill any wild raspberry or blackberry plants, because they can spread disease to your plants.
Set plant in a hole about two inches deeper than the depth to which the daughter red raspberry plant had been growing previously. Use a shovel to tamp down the soil over the top of the newly transplanted red raspberry plant, being sure to cover roots. Snip off two-thirds of the cane with garden shears to help encourage root development. Provide one inch of water. Consider planting at night to help the plant adapt to its new location.