I moved a bunch of summer fruiting raspberries just before they popped their leaves earlier this year, I don't have autumn fruiting ones but a word of raspberry-loving caution. The first year I had one raspberry cane and it sprouted three babies for me. The second year, three turned into eight and the third year's eight doubled or more! I moved them all to a better spot last year and the bare earth subsequently erupted with about twenty baby plants in soil I thought was bare. Turns out my thorough digging wasn't thorough enough. The roots are shallow but somehow I missed them! Anyways, this year I have thirty plus raspberry canes and still some odd roots are sprouting shoots under my veg beds and in between the path and other plants. They really are rampant! So best of luck moving but watch out 