Micronations/Starting your own micronation/Making an Island/Moving a Floating Island
You could have a fleet of tugboats with cables attached to a sturdy part of your island tow it to another location.
If the design is modular the modules can be moved individually, in fact they could be linked in a long chain to minimize drag for travel over long distance and then rearranged on location.
Build a mast near the middle of your island so that you can use sails to drag your island to another location using wind. It's important that your mast is sturdy and large enough to catch enough wind to efficiently move your island, but not so big and heavy that its hard for your island to support it. Also, putting your sails too high might tip your island altogether in rough winds, so it is best to set it such that it is low enough not to let the island blow over (closer to the center of gravity), yet not so low that it gets in the way and/or is not able to catch enough wind. Making the bottom of your island more stream-line would help in reducing drag. This method should be more effective if your island is small.