Large, well-established boxwoods can be transplanted if you can dig out enough of the rootball. With large plants, this is difficult for a homeowner to do without special equipment such as a tree spade, which leaves a large ball of soil intact around the roots. Most of the shrub's roots will be in the top 12 inches of soil, but roots may extend out several times the width of the shrub. It may help to make downward cuts a foot deep into the soil about 18 inches from the main stem one year before you want to transplant them.
Cutting through the roots like that forces them to branch and develop a more extensive root system close to the main trunk. Then the following year you can dig up the shrub and transplant it, attempting to keep as much of the soil in place as possible. The best time to transplant is either late winter just before spring growth begins, or a couple of months before the ground freezes in early fall.