I had a similar problem - if I can recall correctly any tab that was below (after) search results wasn't moveable.

I went around this by setting the page to be a sub-tab of Home (found at Page Settings > Basic Settings > Page Details > Parent Page), or another main tab that was in the beginning of the list.  I then could pull it out to be a main tab, or move up and down.

I went through Gemini looking for this issue but didn't find anything like it.  I had incorrectly assumed it was already reported and was being worked on, so I never actually figured out the bug.

If you can put your finger on the bug - whether it involves the 'Search' page like I had mentioned, simply a matter of how many times you move the page, or anything else, I'd recommend reporting it at support.dotnetnuke.com in the Admin/Host pages section (under Core Framework).