Oh... I completely missed this feature. I thought I had to evacuate them from a storm area... onto another plot of land. My bad!
Viewing post in Stormy Herd jam comments
Yes, you do need to! Some random islands have farmhouses, while others don't. If there are animals on an island without a house, you'll need to pick them up one by one and drop them on an island that does have a house. From there, you can either move them to the house or drop them directly at the door.