Can a father come to the state of the child's residence and just remove him to another state?

If he took the child without notifying your sister, I think that could be considered kidnapping. But to avoid a situation like this in the future, your sister should consult a family law attorney and find out if some kind of custody/visitation agreement can be reached between her and her husband. The agreement would specify which parent had physical custody of the child, when the other parent (not living with the child) could visit the child or take the child on trips, etc. Your sister has the option of filing for divorce or filing for legal separation only. Again, a family law attorney, after reviewing your sister's situation, can determine what your sister's best legal options are.

If a court order is in effect it will control. If, however, there is no court order in effect, than a parent can take their child out-of-state.  However, the other spouse can go to court and obtain an order which will require the out-of-state parent to return to the state.  If at that point they don't return or return but again leave with the child, they can be charged with parental kidnapping.

The safest way for to obtain legal custody of the children is to go to court. What you want to do is get a temporary order from the court on custody and visitation.  It will be temporary because a final decree will be issued upon divorce.  The court will consider the bests interests of the children in making its determination. The court will look at the age of your child, the relationship of the parents and the child, the present living situation of the child, etc. 

Note:  He could file for custody in his state so make sure that your sister files first. Try not to let him take the children until she does.