I do not live in Chatham, but I am quite sure the school district works the same there as it does in my market area. The only criteria for each of the elementary schools is address. In Basking Ridge, if you start the year in one school and you move to another district in the same town they typically let you finish the year where you started - but the next year you have to move your child to the appropriate school in the district.
I hope that helps.
I am an agent in Chatham and I also live in Chatham. The school your child would attend is determined by your address for k through 3rd grade. There are three k-3rd grade schools, one 4th and 5th grade school, one middle school, and one high school. Hope this helps.