In almost EVERY jurisdiction in the country, animals that are allowed to roam out of your yard (mostly cats, as in your case), will be considered nuisance animals if they enter neighboring property - and thus can be trapped and even killed by those property owners, so long as the killing is done in a manner where the animal experiences minimal suffering and the death is quick. Poisoning is a violation because it can contaminate groundwater, and kill/impact other forms of wildlife, like birds, which (with a few exceptions) are protected at the federal level. Shooting or breaking their necks, on the other hand, are generally accepted methods so long as they fall within local rules regarding firearms. Once the cat leaves your yard, it's fair game.
The neighbor and animal control can make police reports. It'll be up to the DA to decide whether or not an actual criminal case will be brought. If such charges ARE brought, and if you have an otherwise clean criminal history, fines and probation are the most likely outcome.

