Swift started out as an animator for Walt Disney Pictures. He worked on such classic animated Disney films as Snow White and the Seven Dwarfs, Pinocchio, Fantasia and Peter Pan. He also worked on The Reluctant Dragon and The Nifty Nineties.
Swift eventually switched careers from animation to screenwriting and made his directing debut with the Disney film Pollyanna, starring Hayley Mills. After Pollyanna, Swift wrote and directed the film The Parent Trap, again starring Mills.
In the late 1970's, Swift returned to Disney and wrote the screenplay for the Jodie Foster movie Candleshoe. He was originally set to direct the picture, before being taken off the project and replaced with Norman Tokar.