This plugin does a great job. I was rather confused at first, and was put off by the initial teething problems reported by people. But it worked - and better than I expected.
Using a child theme, I had some layout and CSS issues and I wondered if this would make it fall over. It did, but I activated the main theme, then activated the child theme, and my layout was returned.
One thing it didn't do was to carry over the sidebars & widgets, but luckily I imported these from the theme on the previous site.
It also didn't update my Slider Pro Plugin settings, the URLs were pointing to images on the previous site. This was a simple case of exporting the Plugin XML, doing a find & replace to insert the new URL, and then I imported the new settings. I just had to update the new Slider ID in the shortcode on the relevant pages.
All in all, this plugin saved me hours of grief. A well earned donatation on it's way, my thanks for a brilliant resource!