APODI, Brazil—The dependable donkey once did it all here in northeast Brazil, from hauling in the harvest to carrying children to remote schoolhouses. Now so many of these ubiquitous beasts of burden populate this vast swath of rural Brazil that they have become a problem—and for some, an opportunity.

Modernity and the skyrocketing sale of motorcycles have demoted the burro from its long-held status. Once cherished here for their hardy load-carrying, donkeys are increasingly seen as a nuisance as they saunter into traffic...