Answer: In my experience it would not be necessary to root prune an azalea because they are shallow rooted and thus relatively easy to dig for transplanting. They are also very sensitive to root disturbances. For best results you would want to take as much of the root ball as possible, thus minimizing the shock to the plant.
If, when you dig it, you are unable to take the majority of the roots, you will probably want to trim back the top accordingly to compensate for the lost roots. If you do not, the plant will likely die back on its own to adjust to the reduced root system.
In addition, it would not be a good idea to move any plant in the beginning of summer as this is probably the most stressful season for transplanting due to the upcoming hot and drier weather. Azaleas are especially prone to struggle during this season, transplanted or not. A better time would be when the plant is dormant, late fall or very early spring as soon as the ground can be worked.
In any case, when you move the plant, water it well the day before, then dig it up (dig as wide as possible because the roots will exceed the branch spread and down about six to nine inches) and place it on a tarp so you can drag it to the new location. The root ball of any established plant is very heavy so plan accordingly.
Replant right away at the same depth as it grew before and then keep it well watered for the next year or two until reestablished just as you would for any newly planted azalea. You will need to keep the soil evenly moist but not sopping wet.
These plants need an acid, humusy, evenly moist yet well drained soil and also benefit from several inches, two to three inches, of organic mulch placed over the root zone year round. (Do not allow mulch to touch the stem however.)
Good luck with the move.
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