This greatly depends on the tree or shrub species in question, and the reason for transplanting. For example, do they need to move out of the way of a construction or utility digging project? Do you want to put them in a spot that better accommodates their mature size, or better meets their needs (sun exposure, soil moisture, deer protection, etc.)? Are you taking them from one property to another due to a move?

If your timing isn't too flexible, such as needing to clear an area or rescue plants that may be damaged by construction, you can move most species just about any time of year that the soil is not frozen or too soggy from recent rains, but how stressed the plants become as a result depends on the size of root ball that can be preserved, and how sensitive that species is with regards to transplanting in general.

Overall, you want to make sure the plant is well-hydrated before the move (not drought-stressed), and the root ball is dug as large as is manageable, given its weight. Most roots will be within the top foot or two of soil, with some shrubs (azalea, rhododendron, blueberry, boxwood) having shallower-growing roots than average. The roots will typically reach further than the branch spread, as long as they had room to grow that far, so a wide root ball is usually more important for preserving fine feeder roots than a deep root ball.

Move plants on a cloudy/overcast day when possible, or consider shading them for about a week post-move, if they are small enough to shield with shade cloth or some other shading material. Monitor them closely for watering needs until they establish; whether from being moved or planted out of a pot, most trees and shrubs take at least 1-2 years after planting to become established enough to need less attention.

If a plant is stressed enough, it can abort any flower buds that have already formed, in addition to wilting leaves or shedding leaves entirely. Leaf drop for plants that are not evergreen isn't necessarily a sign of transplant failure, although this late in the year, new leaves may not grow until spring. If a conifer or broadleaf evergreen loses leaves or they turn brown after being moved, that is a more concerning sign and likely indicates the plant has suffered too much damage.

Spring-blooming species like azalea, rhododendron, lilac, flowering cherry, dogwood, forsythia, and others have already developed next year's flower buds, which lie dormant on the branches all autumn and winter. If they are moved and survive but are too stressed at some point, some of those buds might be killed. (That's ok, but something to be aware of if the plant regrows suitably well in spring but does not bloom. The next year, it will probably flower more normally.)

Summer-blooming species like crapemyrtle and bluebeard do not have flower buds on the branches until just before they flower, so if any branches die back (or need to be trimmed a bit) as a result of the move, they won't necessarily have an interruption in flowering from one year to the next.

Overall, the best time to plant (and transplant) many shrubs and trees is late summer going into early autumn (September). The summer-warmed soil promotes faster root growth, while cooling air temperatures place less stress on the foliage as deciduous plants get ready to shed leaves. Evergreens can also be planted in early autumn, but avoid waiting until late autumn, since that can increase the risk of significant winter damage.

Miri

Thank you, Miri.  I very much appreciate the brief education behind my query.  It helps -a lot.

You're welcome!