What kind of construction, and just what does the "giver" expect to be left there after the building has been moved? And what kind of shape is the building in now?
I have been involved in moving several frame buildings. More or less, we lifted them up off the existing foundation, drove a truck under them, built some structure to hold the building on the truck, and then drove the building and truck to another foundation we had made and set it down. Were any of the projects easy? Not hardly, but we got them done. We never moved a building very far though. And none of the buildings were very large. I would think that a 2 car garage would be do-able with a farm truck.
Removing the foundation and concrete floor if there is one is a different problem, and one that would require some heavy work, probably by big machines and a dump truck. If that is part of the deal for the "free" garage, I would doubt that the owner would get any takers.
But if the building is in nice shape, but for one reason or another is in the way, they might find someone who wants it and can and will move it.
There are companies that move buildings all the time. They have the expertise and the equipment to do the moves quickly, safely and usually are not nearly as expensive as having an equivalent building built. I would suggest at least checking out having the building moved.
There probably should be a contract that spells out what the current owner expects the person who gets the building to do, and who has the potential liability for anything that might go wrong.
Or maybe no one will be interested in the "free" building and if the owner really wants it gone, they might need to tear it down or pay someone else to remove it.
I would sure try to give it away. Good luck!