My mother has had temporary custody of my now 4/5 year old niece and nephew. It has been over 4 years in and out of court. Two years ago she finally decided to separate from my biological father. He was very abusive to us and has a criminal history of drug possession, trafficking, and assaulting a police officer. He spent over 10 years in jail since i was born. Three of my siblings were shot because of his lifestyle and are now following in his footsteps. My father and 2 of my siblings moved with him to another state with his father who is also a criminal/registered sex offender. My mother got an order of protection against him (which has been violated) and he's trying to use the children as a way to get to her. He has done this with me and my other siblings. There are more details.
Child Custody Lawyer
Joint custody is out of the question since he and your mother are separated and not on good terms. Further, he lives in another State. As for visitation, he doesn't really sound like a candidate for it. If his lifestyle is as you've described, I don't see a court granting him visitation.
Criminal Defense Attorney
If the parents are alive and not unfit as parents, the court would not give the children to a grandparent for custody, especially if abusive and unfit. In fact, visitation to such a grandparent would be a stretch. Consult a family court attorney.
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No. In New York, grandparents have no standing to sue for custody unless the biological parents are dead or deemed unfit by a court. All the details you provide would have to end up in an order finding these people unfit for another family member down the line to get custody.