Answer: You can try almost anything in gardening and sometimes it works, sometimes it doesn't. Since the alternative is to leave them behind (is there any chance the new residents would share some berries with you?), you might try potting up a few in large containers and moving them that way. A good time to dig them up is while they are still dormant. Try to take as much root as you can and use the biggest containers you can handle. It would be less stressful on the plants to trim them back hard the way they were when you got them, so you might want to do that to some of the plants but also leave some untrimmed to try to keep some to harvest. Take good care of them in the containers and replant as soon as you can.
For the plants you will try to harvest from, remember to prune if you haven't already. You can prune the canes that have produced fruit down to the ground immediately after harvest. In the very early spring select the thickest remaining canes and cut them back to about 7'. Remove the spindly canes completely, and prune back the side branches to direct growth into new, flowering stems near the tops of the remaining canes. Following this pattern should provide healthy, productive canes year after year.
Good luck with the move!
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