Raspberries are best transplanted in the spring. If you want to transplant your raspberry bushes into containers or raised beds, then wait until late summer or early fall to do so. These plants produce a lot of fruit at this time and can easily become overcrowded with too many berries. Wait until after they have stopped producing new growth before planting them outdoors again. This will ensure that there is enough space for their roots to grow without being crowded by other plants.