ABU DHABI, United Arab Emirates—The 30 beauty-pageant contestants stood still. Five judges looked them up and down, appraising their facial beauty, leg length and chest size.

None of these contestants, however, were likely to break into song or twirl a baton. They were birds of prey, tethered by their legs to the judges at the second annual World’s Falcons Beauty Contest. Victors at the two-day pageant, held in this capital city earlier this month, received prizes of 10,000 U.A.E. dirhams, or $2,700.

Falcons...