I won't get into extracting tools for this post, site specific and everyone has favourites.
Material for packing the rootball though, extremely important! The roots need to be kept moist of course and immobilized very well, keeps soil on, reduces mechanical damage etc..
Lots of water in the pack and a misting bottle to frequently spray roots that are exposed during to extraction process.
Sheets of pre-moistened material and the stretch wrap that Pinion mentioned!!!
These items above are pretty much the most important items in my pack, I have forgotten to pack this or that often enough when collecting before. If I don't have my mister, stretch wrap or fabric I really feel like an idiot with reduced chances, I'd rather forget the shovel, seriously.
Thin lightweight fabric like an old bedsheet or light burlap is good, coarse burlap is heavy and bulky. Scissors to cut sheets to size are nice.
Get the rootball atop the fabric, add folds/bunches of the fabric where needed for support and padding, fire some moist moss in here and there if needed.
Treat it like a first aid injury, immobilize the hell out of it with lots off padding in voids and pack'er tight! Cover the whole thing tightly with the stretch packing wrap, there's a lot of wrap on one little roll.
I spent plenty of time packing with just string, tape, plastic bags and moss like we see photos of even highly regarded collectors still doing nowadays. Never again.