You can transplant blackberries in either spring or fall in Missouri, though the Cooperative Extension Service at the University of Missouri recommends doing it while plants are dormant--that means transplanting by March, at latest, in most areas of Missouri. Whether you transplant in the fall or spring, add a leaf or straw mulch 2 to 3 inches deep to protect the roots from frost heave during the cold months. This is especially important for fall plantings.
Cut back the canes to around 6 to 10 inches in the fall before transplanting. Even if you plan to do the actual transplanting in spring, the plant will suffer less transplant shock if you give it time to rest after cutting the canes.
Loosen the soil carefully all around the blackberry bush you want to transplant, using a garden fork. Lift and pry to expose the roots without breaking them.
Brush away loose dirt as you slowly but firmly pull the plant out of the soil with your hands. It will be impossible to avoid some breakage, but blackberries are tough plants to kill, so do not worry about minor damage.
Place the plant immediately in a pail of water to prevent drying out the small roots.
Dig a hole as wide and deep as the root system of the plant. It should sit at the same level it was in its previous location with plenty of room to spread roots out without crowding.
Hold the plant in place and carefully crumble loose dirt around the roots, pressing it down firmly to make good contact with roots. Continue adding soil until the hole is full and level with the surrounding soil. Do a final firming of the entire excavation by pressing down with your foot all around so the cane is upright and secure.
Water the cane thoroughly to help eliminate air pockets around fine rootlets and hydrate them. Use enough water to saturate the hole to the depth of the roots, and keep moist until the plant is fully established.