well there are many factors. It depends on the age of the plants - how long it has been growing in that area. It depends on the kind of plant. Obviously digging out old plants would be traumatic for them. You need to try to dig out a large ball of root to keep it undisturbed as possible.

For vines - i would first cut down the vine to a manageable size. (the less foliage & extra branches, the higher the survival rate) Second I would do the transplant only when the plants have gone dormant which is sometime in the fall so when you do uproot the plants, they are least susceptible to transplant shock. Transfer them to their next location and make sure no air pockets remain in the soil otherwise frost can kill the plants. Top with mulch to protect them from the worst of winter. this can also be done in the spring when the plants are just coming out of dormancy.