Updating website design – workflow?

Hi,
I am trying to redesign a client website that I have inherited.
I have imported it to my multisite installation (a task that I have found non-trivial whenever I have attempted it – wordpress export and import seldom gets everything!), I have cloned it using NS Cloner 3 and spent some weeks changing and customising the theme, redesigning the menu layouts, etc.
I would now like to find a way to import any updated or new posts, pages, events etc. from the original site so I can make this new one the live site.
Does anyone have any idea how to do this? I could clone a blank copy of the site (using NS Cloner) and then reimport from the old one, but as I said, this always leaves stuff behind in my experience, especially extra data from plugins (such as events calendar). Surely someone must have come up with a workable method?

Any insights would be greatly appreciated!
Thanks