In the film, Duke and Turk were originally the captain's dogs until the ship the Robinsons were on was chased by Kuala and his pirates into a storm and found itself stuck on rocks which weren't too far from land.
Francis Robinson, the youngest of the family, insisted they bring Duke and Turk with them to the island, despite Father Robinson's fears that they might sink the raft they made and sailed on. But fortunately, they don't sink the raft when Father finally allows them to climb in.
The following day, Duke and Turk help Mother Robinson search for Francis, who had found a baby elephant he named Rocky, while at the same time fending off the tiger that was on the island.
On the day of the Robinsons' holiday, Turk was one of the animals who participated in the race with Francis' monkey riding him, along with Rocky the elephant (who had Francis riding him), Lightning the mule (who had Fritz riding him), Clementine the ostrich (who had Ernst riding her), and Duchess the zebra (who had Roberta riding her), while Duke stood with Mother and Father Robinson.
During the race, after Duchess lead Roberta to the beach where Roberta spotted the pirates coming ashore to look for her since she was their former prisoner mistaken for a boy at first, Duke and Turk were seen running back and forth while chasing Duchess; probably trying to bring her back to the family. Then, Duke and Turk were later called by Francis with a whistle, which gave away the family's whereabouts to the pirates, and joined the family on top of a hill where a battle against the pirates was about to occur.
At the end, after the pirates' deaths and as Ernst travels to London with Roberta's grandfather, Captain Moreland to continue his education there, Duke and Turk were last seen running away along with Clementine, Duchess, and Rocky as Francis tried to bring them back.