Do I have the right to move a custody case from family court to supreme court? How do I get the case moved?

If there is a divorce case pending in Supreme Court you can file a motion to consolidate and have the family court case consolidated with the Supreme Court case. If you have not yet filed a Supreme Court case (seeking a divorce) then you can do that and move for the family court case to be consolidated with your new Supreme Court Case. I should caution you that if the family court has begun a trial or completed a substantive amount of work on the case (ie: you, your husband and child(ren) have met with a forensic evaluator, your request to consolidate this family court case may be denied. In any event, I suggest that you speak with your attorney about your concerns so that you can come up with a plan.