What you're seeing is expected behavior. I was worried that the geometry itself was somehow different in the exported model. However, the difference you're seeing is that the entire part is in a different location than in the design it was exported from. The reason for this is that you are exporting an occurrence (an instance of a component). Occurrences can be repositioned within an assembly and you can even have multiple occurrences that reference the same component. When you export a body that you select in an occurrence, the body that's exported is the body that exists in the component (which you don't actually see) and it uses the coordinate system of that component.
There's a description of this in this section of the API help. Hopefully, that will help.
http://help.autodesk.com/view/fusion360/ENU/?guid=GUID-88A4DB43-CFDD-4CFF-B124-7EE67915A07A
The only time when you can always expect the exported body to be in the same position as the original body is when the body exists in the root component of the assembly. It's then exported relative to the assembly coordinate system because that is the same as the root component's coordinate system.
If you need the exported body to be in the same position as it was in the assembly, you can work around the behavior by copying the body from the occurrence into the root component. This is fairly easily done by having the root component (main assembly) active, select the body you want to export. To make sure you're selecting a body and not a component it's best to select it in the browser. Right-click and pick "Copy". Right-click in an open area of the graphics window and click "Paste". Click "OK" on the "MOVE/COPY" dialog to leave the copy in the default location. Now, export that copied body, which will be in the Bodies folder of the top assembly. You can undo to get back to the original state.
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Brian EkinsInventor and Fusion 360 API Expert
Website/Blog:
https://EkinsSolutions.com