If the domain+tld is the same and only design has changed, I would try my best to preserve old links to posts.
I assume, most heavy part of the site is the blog posts and other templates / pages are less numerous, so I would use default Couch Migration tool to export old content and would migrate it to the new site as is. In the admin of the new site then I would simply delete templates / pages that no longer exist in the new site and add missing templates from the new site.
This is a quick way to 'migrate' old posts, keep them links and preserve everything that was working previously. So blog posts would stay, and pages like contact-us, about-us, etc can be added with much less effort. I have used Migration a few times and it worked flawlessly, so my idea of using it to move blog post should work as expected (don't forget to move images keeping the dir structure in /uploads/ as the relevant links would stay the same).
For complex cases, it is possible to make an export-import utility but so far it was not requested. Anyways, CSV Exporter-Importer is available for public, but would require some advanced tinkering to correctly export rich text posts.