Live capture and relocation of deer is seldom practical unless public relations mandate live removal.
Translocation of deer is very expensive and can lead to death of the deer due to capture myopathy. Most states currently do not allow translocation of deer.
The most convenient method of euthanasia is shooting. Placement of the shot is critical to ensure rapid death.
Check your state regulations regarding disposal of carcasses. Some states may have specific instructions if deer have been harvested in areas with chronic wasting disease (CWD).