When is the best time to transplant a hydrangea and should they be cut back in the fall?
While they can transplant fairly easy, make sure it is done while dormant and get as much of the root system as possible. This article should also help: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/growing-hydrangeas-hydrangea-care-guide.htm
Typically, pruning it back does help when transplanting but if you are planning on moving it during fall, I would not take off too much, as the shrub may need its foliage for additional protection in winter (if you're in a colder climate). This article will help with pruning: https://www.gardeningknowhow.com/ornamental/shrubs/hydrangea/prune-hydrangea-bushes-hydrangea-pruning-instructions.htm