LOS ANGELES — Walk the halls of Netflix Animation, spread across three buildings in the heart of Hollywood, and a cheeky question may cross your mind: Is anyone left at Disney Channel headquarters?

Chris Nee, the force behind “Doc McStuffins,” the groundbreaking Disney Channel series, decamped for Netflix in December. She followed Alex Hirsch, a cartoon whiz who created “Gravity Falls” for Disney. Naketha Mattocks, a former executive at the Disney Channel, where she helped steer the popular “Descendants” movies, now heads up Netflix’s family film unit. She is working with Kenny Ortega, a veritable Disney legend known for shepherding the blockbuster “High School Musical” franchise.

“That he has chosen to make Netflix his creative home to work on both feature films and series is thrilling,” Netflix’s chief content officer, Ted Sarandos, crowed in April about Mr. Ortega.

Netflix has been a force in children’s entertainment for years, its usefulness as a digital babysitter helping it grow into a streaming behemoth with 152 million subscribers worldwide. But it now faces the biggest competitive threat in its history: Disney, the company that has supplied much of Netflix’s most popular programming for children and families, is determined to be a streaming giant in its own right.