How do I move a property into a different "directory"?

I think you mean moving an object or instance into a different "directory" in your title, not a property.

Nevertheless, to move an object/instance, you can simply reassign its Parent property to the new location you want it to be in. In your case, you want to move a loading GUI into the StarterGui, however, another mistake that I have found is that you should not be parenting objects to StarterGui during the runtime of the game. StarterGui is only replicated to a player's PlayerGui when he/she joins and each time he/she respawns, thus any changes you make to it won't be applied until the next respawn time.

That is why I recommend you parent your loading GUI directly to the player's PlayerGui. For example:

Module.LoadingGui.Parent = player.PlayerGui