Oooh, you lucky thing!
If they are fruiting so abundantly now, they are clearly happy where they are! I would think carefully about whether you really need to move them to another spot. I just took some rasps that were doing exceptionally well in my garden to my allotment, and they have taken a big hit on being moved to a new site. I've almost no fruit this year. While they will recover and fruit again next year, it's not really worth shifting them unless there is a pressing reason to move them.
Instead, think about whether you can tidy them up in situ, removing a few plants so that you have a couple of clear lines, and tying them in so that you have neat rows. Rasps produce baby plants (suckers) all over the place, so you can use the healthiest ones of these to replace any weak older canes. Once you've organised them a bit, you should then be able to weed them much more easily. A lot of people recommend a really deep mulch for fruit to suppress weeds and lock in moisture - if you have a brewery near you, you may be able to pick up spent hops for free and put them down next spring.
In terms of moving things, I absolutely definitely would not move anything over the summer, and certainly not in a heatwave! Apart from the fact that you will be frazzled, the plants will really struggle to get the water they need to repair themselves. Leave moving of all fruit plants til November at the earliest, and get them in place before the end of March. (This includes gooseberries etc).
As Bowsaw said above, different kinds of raspberries need different pruning strategies. This sounds complicated, but it really isn't! With summer fruiting rasps (i.e. those fruiting now), once they've finished, you cut down the canes that have flowered and fruited to the ground and leave the others. It's usually really obvious which is which because they're different colours - the old stems are a browny/grey colour while the new ones are green. Autumn fruiting canes (which fruit in September) get chopped down to the ground after they fruit - you have to steel yourself a bit, because it feels really brutal, but it does work.
In short, I would do the pruning of the rasps just after they have finished giving you loads of lovely berries, but leave any moving of any plants to between November and March.