MEXICO CITY—As a shortage of lifesaving drugs to treat his 5-year-old daughter’s leukemia intensified, Israel Rivas decided it was time to take drastic measures. Last month, he and dozens of desperate parents in similar situations blocked the main entrance to the domestic terminal of Mexico City’s airport.

“If there are no medications and your child is fighting for her life, what does a parent not do?” said Mr. Rivas, who was prevented by airport security from carrying out his goal of lying down on the runway.

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