Unmarried couple with kids part ways. Mom takes one of two kids and moves out of state against Fathers wishes. Can any criminal charges be brought against Mom? Can she move prior to any custody and or visitation agreement is made? Or what if Mom only takes one child out of state and leaves the other sibling with Father? Is this abandonment ?

First its not a good idea to piggy back your question on another thread. To avoid confusion, I've moved your post to its own thread.

Second, Any question on law needs to include your general locale as laws vary by area.

Until there are court orders governing custody and visitation each parent has equal rights. So each parent can take either or both of the children. However, until the moving parent establishes residence the other parent can file for custody and possibly force the child to return.

A parent that take a child away from the other parent and hides that child from the other parent may be at risk for prosecution for parental kidnapping.

A parent that leaves a child with the other parent and moves away without contact, may be considered to have abandoned the child. But this would not be a criminal charge (except in a few areas) but could be used to get full custody.