If the islands are close enough together, most mammal species that humans ride might be able to swim from island to island. I once read that most land mammals, except for primates, are strong natural swimmers.
Horses swim. Donkeys swim. Mules swim. Camels swim. Elephants swim.
I note that humans swimmers often attach swim fins to improve their swimming, and swim fins could be made to attach to the limbs of various land mammals to improve their swimming.
One or more trained cetaceans might pull a boat by holding in their mouths a bar attacked by ropes or chains to the boat. One or more trained cetaceans might put their heads to the back of the boat and push it across the ocean.
I guess that teams of trained land mammals might also be able to pull or push a boat from island to island if the distance is small enough.
And of course a small boat might be built like an elephant's howdah and strapped to the back of a large cetacean for a trip between islands.