Mr. Jiménez’s benchmark case exposes a little-known but apparently widespread practice. Many American hospitals are taking it upon themselves to repatriate seriously injured or ill immigrants because they cannot find nursing homes willing to accept them without insurance. Medicaid does not cover long-term care for illegal immigrants, or for newly arrived legal immigrants, creating a quandary for hospitals, which are obligated by federal regulation to arrange post-hospital care for patients who need it.

American immigration authorities play no role in these private repatriations, carried out by ambulance, air ambulance and commercial plane. Most hospitals say that they do not conduct cross-border transfers until patients are medically stable and that they arrange to deliver them into a physician’s care in their homeland. But the hospitals are operating in a void, without governmental assistance or oversight, leaving ample room for legal and ethical transgressions on both sides of the border.

Indeed, some advocates for immigrants see these repatriations as a kind of international patient dumping, with ambulances taking patients in the wrong direction, away from first-world hospitals to less-adequate care, if any.

“Repatriation is pretty much a death sentence in some of these cases,” said Dr. Steven Larson, an expert on migrant health and an emergency room physician at the Hospital of the University of Pennsylvania. “I’ve seen patients bundled onto the plane and out of the country, and once that person is out of sight, he’s out of mind.”