RIO DE JANEIRO — American parents involved in international custody fights in Brazil have managed to get presidents, secretaries of state and lawmakers to help settle high-profile disputes that have long been a thorn in bilateral relations.

In the latest case, Chris Brann, a Houston physician, has found a unique way to gain leverage over his former spouse, enlisting federal prosecutors to step in.

On Wednesday, F.B.I. agents arrested his ex-wife’s parents when they landed in Miami and charged them with international parental abduction. In a tactic that experts in international custody fights described as unprecedented, Mr. Brann then signaled that the criminal case against them might be dismissed if his ex-wife were to return their son, Nicolas, to the United States immediately.

“Ultimately, I felt like this was the last hope,” Mr. Brann said in a telephone interview on Wednesday afternoon. “What’s in my child’s best interest is to have both parents in his life.”