Relocation of coyotes is suitable only for rescues.
Translocation of coyotes is not practical in most situations, nor legal in some states.
A .22 rim fire short or CB cap to the head is a good method for euthanasia of coyotes if allowed in your municipality. Do not shoot a coyote in the head if it must be tested for rabies. If you are authorized to use immobilization or tranquilizer drugs, the animal can be sedated and moved to another location to be euthanized. Immobilization and transportation is useful where shooting is prohibited or indiscreet. Carbon-dioxide also is an appropriate method. Use a snare-pole to control the coyote. Transfer the coyote from the trap into a suitable cage.
In public areas, transport the carcass in a large trash bag to avoid questions from the public. Check your local and state regulations regarding disposal of carcasses.