Updated April 16, 2020: A parent expressed concerns yesterday about issuance of a passport for a child. The parent's fear was that the other parent, having no right of custody, might be able to obtain a passport for the child and to remove the child from the country. Certainly, the Hague Convention, can be of assistance in recovering children who are wrongfully removed. However, this is an expensive, time-consuming, and not always successful process. This is particularly true if a child is taken to a country that is not a Hague Convention signatory or to a country that is a treaty partner but which is non-compliant.
As a practical matter, Homeland Security is now requiring that any child have his/her own passport -- even for travel to Canada or Mexico. In a high conflict case involving one or more foreign-born parents, where there is a potential flight risk, there are certain protective measures that will help to prevent a parental abduction.
One situation that is encountered more frequently recently, since many parents are not married to each other, is that one of the parents may remove the child from the country (if this is an international abduction) or out of the state. In Michigan--a very large state--one abduction with which I am familiar involved a father removing a toddler from the Upper Peninsula to an area in upper Michigan (in "the mitten." The distance between the mother's home and his home was about 350 miles. This means the abduction might as well have been to another state given the travel time. It's been my experience that usually, the "taking parent" is the mother, but this is not always true. I've handled international and interstate abductions where the father was the "taking parent."
When the parents are not married to each other, but they have executed an Acknowledgment of Parentage, that document, consistent with The Acknowledgment of Parentage Act of 1996 is clear about which parent has the right to custody of the child: