Albert Einstein became one of the original poster children, literally, for refugee resettlement after he was banned, along with all Jews, from academic work in Germany. Mikhail Baryshnikov and other Russian ballet dancers followed decades later when they were famously offered protection from Soviet oppression in the United States and Canada.

These cases were clear-cut in the public consciousness: These people were largely fleeing government persecution based on their race, religion, nationality or political views — four discrete categories for asylum under international law — and Republicans and Democrats mostly agreed on helping them.

That consensus has dissolved more recently with the expansion of a fifth, much murkier, category of refugees, which has ballooned to protect victims of modern threats coming from nongovernmental entities such as gangs and terrorist organizations. Recently, claims have also been recognized from gay and transgender people in countries that persecute them and female victims of domestic violence in places where the government refuses to protect them.

President Trump’s administration is in the midst of a targeted push to turn back the clock.

“Asylum is for people fleeing persecution, not those searching for a better job, yet our broken system, with its debilitating court rulings, a crushing backlog and gaping loopholes, allows illegal migrants to get into our country anyway, and for whatever reason they want,” Kirstjen Nielsen, the homeland security secretary, told Congress in May. “This scamming of the system is unacceptable.”

More than 700,000 cases are now pending in the American immigration courts, with the largest numbers of people waiting from Mexico and Central America. The growing backlog, plus a shortage of judges, has steadily increased the wait time for asylum and other types of immigration cases, which now take nearly two years on average to complete, according to data maintained by Syracuse University.

One of the biggest blows to asylum petitions came last month, when Mr. Sessions overruled the case of Aminta Cifuentes, a Guatemalan woman who suffered a decade of abuse by her husband that included acid burns and punches to her stomach when she was eight months pregnant. Her baby was born prematurely and came into the world with bruises.

Ms. Cifuentes’s lawyers argued that she was the victim of a larger crisis facing women in Central America. They presented a slew of reports by human rights groups as evidence that gender-based violence was a systemic problem that had been ignored by the police, and convinced a panel of judges that the existence of “a culture of machismo and family violence” in Guatemala meant that such women should be considered, for purposes of asylum claims, their own “particular social group.”