If both parents are in the military, living together for several years can one of the parents (mother) request a transfer and remove the child from that state and the other parent? The child's name on the birth certificate is that of the father.

Is there a custody order?

The name on the birth certificate has no bearing, If the father is listed on the birth certificate it merely saves him having to prove paternity.

If there is a custody order, the rules or agreement there will decide. If there is no order the father in this case can file for an emergancy hearing to order the child be returned to his state. But with that said, it can also tie him into not being able to move. The mother in this case can argue in court that being in the military will cause both of them to have to move and that in time the father will be forced to move also.

I had one of these cases some years back and the judge actually awarded custody to the grandparent who lived in that state, ( it had grandparents right laws) but I understand that case was strange and not the norm.

But the father in this case may attempt to keep the children from moving out of state.