depending on the complexity of your design, this can be a chore or a CHORE.
You can enter layout mode on a given layout, select all objects, copy to the clipboard, switch to the new layout in the other file and paste. The resulting layout may be perfect or need a lot of fixing after the paste.
You can use the inspector to set the height of each layout part in the new layout to match the sizes in the original. You can use the inspector to make sure the top, left position of your group of pasted objects is exactly the same as your original.
You can also import scripts and given the way scripts reference layouts and layouts reference scripts, you may want to copy and paste a layout's objects, import the scripts, then delete all the layout objects and paste them again--that preserves references in both the scripts and the layouts.
There's also a third party product called FMMigrator that some have recommended.