MIAMI — The donations kept pouring in: hundreds of thousands of dollars in campaign contributions to President Obama and more than a dozen members of Congress, carefully routed through the families of two wealthy brothers in Florida.

They had good reason to be generous. The two men, Roberto and William Isaias, are fugitives from Ecuador, which has angrily pressed Washington to turn them over, to no avail. A year after their relatives gave $90,000 to help re-elect Mr. Obama, the administration rejected Ecuador’s extradition request for the men, fueling accusations that such donations were helping to keep the brothers and their families safely on American soil.

“The Isaias brothers fled to Miami not to live off their work, something just, but to buy themselves more mansions and Rolls-Royces and to finance American political campaigns,” President Rafael Correa of Ecuador told reporters last month. “That’s what has given them protection,” he added, an allegation the Obama administration and members of Congress reject.

The tug of war over the Isaias brothers, sentenced in an embezzlement case, is part of a broader battle between Ecuador and the White House over international fugitives, including two other men the Obama administration would like to get its hands on: Julian Assange, the founder of WikiLeaks, and Edward J. Snowden, the former intelligence contractor.