For evergreens it is best to transplant them while they are actively growing. Spring is a good time but I've had great success in the middle of summer too as long as I keep them watered.
But for trees that go dormant, it is best to transplant them when they are dormant. The roots will grow some but the tree itself does not have a lot of sap running in it. If you damage the tree by breaking branches or scraping the bark and the sap is flowing, and you probably will, the tree can lose a lot of energy by that sap leaking out. Diseases are normally more active in warmer weather too and an injury gives them an opening to get inside the tree. A tree transplanted when the sap is running is likely to wilt and be weakened. Insects seem to be attracted to weaker plants. I've transplanted a lot of deciduous trees and bushes when the sap is flowing and they usually do well, but the odds are a little more in your favor if you can transplant it when it is dormant.
I've pruned trees when the sap is flowing when I need to, either because of damage or to remove sprouts growing below the graft line. It's best to prune when the tree is dormant (other than certain flowering bushes which may need to be pruned before next years blossoms are set) but it is not a case of absolute success one way and absolute failure the other. It's just a way to improve your odds.