Vegetable plants pulled from the ground go into transplant shock the moment their roots are exposed, so speed and moisture matter more than anything else. Get them into flats with damp soil around the roots and out of direct sun as fast as you can.
Pro tip: Label each flat with the plant name and planting depth so you replant tomatoes and peppers at the correct depth, not just wherever there is room.
Watch out: Roots left exposed to open air for more than a few minutes dry out and die back, weakening the whole plant.