They have spent thousands of dollars in fees, scaled a mountain of paperwork and waited nearly five years, but Caroline and Jason Lankford have yet to be matched with the Ethiopian child they long to adopt.

They are among thousands of Americans whose hopes of becoming parents through overseas adoption are on hold as the number of eligible children dwindles. Overseas adoptions by Americans have dipped to the lowest level in 35 years, data released on Wednesday showed. The State Department reported that it issued 5,372 visas to children who were adopted abroad or were coming to the United States to be adopted by American parents in 2016, down from 5,648 in 2015 — and a fraction of the 22,884 overseas adoptions in 2004, the peak.

“After all this time, we don’t know when we’ll get our boy or girl,” said Ms. Lankford, of Anchorage, who sought a child from Ethiopia because her husband, Jason, had worked for a time in Africa. “We knew so many kids wanted a home.”

The latest figure is the lowest since 1981, when there were 4,868 overseas adoptions. Reasons for the decline are varied. Some countries are promoting domestic adoptions over foreign ones; several have suspended the process because of corruption. Still others have imposed stringent restrictions after cases of child transfers and abandonment.