I wouldn't transplant. I would cut some and root them.

What a bonanza you have there!

Caneberries include raspberries and blackberries. An individual raspberry stalk is commonly referred to as a cane. Blueberries grow on bushes.

Moving/transplanting these during the hot, dry season will likely end in failure. Maybe you can wait at least until harvest is over.

Perhaps another forum member has a method.

Are you sure you have raspberries and not blackberries? Blackberries are better suited to your climate, especially since you note that it is typically hot and dry during the summers. Wild blackberries are common along fence rows in most of the South. Just confirming.

In your situation, I would manage the thicket as best I could during the summer, and do the transplanting in September or even October, when it is cooler, and the plants are beginning to slow down their growth. You can pick the plants with the best fruit, and tie a string or something colorful around the base, so that you know the best plants to dig up and move. With raspberries, three feet spacing between clumps works pretty well. Most people plant them in a row, so there is access from both sides. If they are healthy and vigorous, they will put our runners to either side, and it will be necessary to pull out the volunteers once or twice a year, typically in the spring and again in the fall. These volunteers can be dug up and used to establish new raspberry beds, if you know of other people who could put them to use. Otherwise, they are simply weeds, and they go on the compost pile. If you do not actively manage the bed, it will quickly grow into an impenetrable thicket.

Now is not the time to move them. You might be asking around for people who want some. You will have extras. Basically I would plan a whole back yard makeover and do it in the fall or early spring.

As for containment, they like to spread. I am thinking of buying sheet metal (yellow pages for a sheet metal shop) 12" depth would contain pretty well, but over time not 100%. As they pop up in your grass, normal mowing takes care of that. That is how my father in law contains his and he is fussy about his lawn.

Agree,..don't cut now,..sit back and evaluate over a season or
two,..you might regret in what you did.
Find out when they fruit of what you have and what you actually like or dislike,..can't find out when all is cut.

Sounds like they weren't kept up, you can cut any old brown canes this summer - they should be starting to fruit down there now since mine are blooming up here. Don't cut the ones that are fruiting now until after harvest, and don't cut the young green ones, but you can cut the old brown ones that have fruited in the past now, and after harvest cut this year's fruiting canes down to the ground.

Then all you'll have left is this year's canes that you can transplant.