If you have customised the new button order and/or default content type for a list or library, then expect these changes to be lost if that site is moved. SharePoint Bug! You may have done this in order to change the content types that appear under a list’s new button or change the content type that used by default (i.e. when a user just clicks the new document icon rather than selecting a content type from the drop down).
As you can see in the image above, I have a library configured with a restricted set of content types available under the new button. After moving the site (using Site Settings > Content and Structure) these customisations are lost. See the next image.
So that’s the issue but what can you do to fix the situation if moving sites is something that you need to support regularly? Firstly, this isn’t the only issue you may encounter.
With the luxury of a farm solution this can fixed using a web event receiver. Using the WebMoving event you can store list new button order information (e.g. in a web property bag) and then in the WebMoved event, this information can be read and applied. I don’t have a code example of this as in my situation it was suitable to apply a static new button order to lists based on the site definition and list template.
If you are unable to deploy a farm solution (e.g. SharePoint Online) then this issue is more difficult to solve. I assume that you can write remote event receivers for web events and apply the same logic as above (I haven’t tried this). Otherwise I can’t think of another reasonable solution beyond writing a control with client JavaScript that ensures the new button order upon viewing the page. However this would require knowledge of what the new button order needs to be, and hence would be more complex to implement if users are free to change it.