Mr. Minter also worries that the military may seize on unrelated medical issues as a pretext for rejecting transgender recruits.

One applicant in Ohio spent five months submitting more and more medical records, and then was rejected in late May because of knee surgery he had as an infant. The applicant, who asked not to be named because he still hopes to join the military, said he was dumbfounded at the rejection, because he has had no issues stemming from the surgery for 25 years.

The Defense Department declined to make any officials available for interview, citing pending litigation. It refused to say how long recruits have been kept waiting or how many have been rejected on medical grounds. But it said in a written statement that it “continues to comply with the court order,” and that “the time it takes to review each individual record will vary based upon the individual.”

Thousands of transgender troops, who officially came out or transitioned in the military when the Obama administration decided in 2016 to lift a ban, are serving now. A RAND Corporation study in 2016 estimated their number at between 2,000 and 11,000. Many are in demanding jobs and have deployed overseas.

Leaders of the Army, Marines, Air Force, Navy and Coast Guard told Congress this spring that they have seen no issues with the transgender troops. “As long as they can meet the standard of what their particular occupation was, I think we’ll move forward,” Gen. Robert Neller, the commandant of the Marine Corps, said in his testimony.

But the Trump administration continues to oppose any transgender military service. Before it was blocked by the court injunctions, the administration sought not only to keep transgender troops from joining, but to discharge those already in the ranks. Defense Secretary Jim Mattis issued a memo in February saying their presence threatened to “undermine readiness, disrupt unit cohesion, and impose an unreasonable burden on the military.”

Last month, the Justice Department filed a motion to overturn one of the injunctions, arguing that the panel of Defense Department experts who created the Trump administration policy had the necessary authority to ban particular categories of recruits, and that the court had “provided scant explanation for disregarding that reasoned and reasonable military assessment.”