In almost all areas it is illegal to transport live trapped animals for release on any property other than your own, unless you have the land owner's permission.  Transporting a varmint, and releasing it where it can become someone else's problem is IMO irresponsible. 
1.  The varmint is being introduced into new territory.  Existing varmints will not take kindly to the visitor, and will most likely try to kill it.  The varmint, if he does succeed in becoming established will have to find new housing, new feed sources.
2.  The varmint has already become habituated to seeking food at the expense of human livestock or other food sources provided by the human.  He is now trap wary.  He will be even more of a problem in his new habitat.
3.  While you may think you are releasing it far from habitation, the truth of the matter is this:  If you were able to drive on a road to get within a mile of where you released the varmint, you are in some one else's neighborhood.  I personally would be infuriated if some one drove "out in the country" to release a problem varmint that then ended up in my yard.
Do the responsible thing.  If you trap it, you kill it.  If you don't want to kill it, then don't trap it. 
Make your coop/run predator proof with appropriate skirt, well attached welded hardware cloth over all openings greater than 1", cover over top of run.  Shut your birds into secure coop every night if your run is not predator proof.  A coon can breach any latch that a 2 year old child can open.  Electric fencing is a wonderful thing!