In order for him to get custody, there would need to be a substantial change in circumstances since the last court order. Custody is based upon the best interests of the child in NY. Based upon what you have asserted here, in all likelihood, he would probably not be granted a change in custody. If he files for a modification, seek an attorney.
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Ms. Blizzard-McGrath has provided an excellent answer with which I agree. He can certainly try to get full physical custody, but that does not sound like it is "the best interests of the child". I also agree that if a petition is filed in Family Court, you should seek representation whether or not your ex- also does. You can find many good Family Court attorneys on this website who appear in Suffolk County Family Court on this website using the "Find a Lawyer" tab. Most provide free initial consultations.
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I agree with the prior responders. The court will not award custody solely because someone has a townhouse. Get an experienced family law attorney to assist you.
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In order for him to get custody, he would have to allege and establish a change in circumstances warranting a change in the custodial schedule. Courts will do what is in the best interest of the child. The fact that he bought a house and lives with his cousin and and her 8 year old, means nothing.
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