Welcome to Permies Sandra!
Spring time has been the best time for me to transplant blackberries. Most other things seem to do better when transplanted in the Fall here, but the blackberries have done best when transplanted in the Spring. I bet they would be fine to be pruned and transplanted, they seem pretty tough.
I got tired of the wild blackberries cutting me to shreads as well.

So I got a mix of the thornless ones last year and have been really pleased with them so far. They have great flavor and are actually more healthy and vigorous than even the wild blackberries!
Hope your blackberries continue to do well!
-Steve