When I'm in a Finder window and enter terms in the spotlight search box (top right of window) and then press the TAB key the search location bar appears just below and the focus moves to the first location (usually "This Mac") shown (as it should). But then the focus seems locked in place and no keyboard key or combination I've found will get it to move (most of the time, that is; at times it's possible to select one of the shown locations with the keyboard and sometimes I've found it possible to tab to the Save button or Add Option [+] button and on to the options themselves but most times it just gets stuck [like now!] and I need to use the mouse). I've been annoyed by this for years now and am running out of new key combinations to try! Is there a way to continue with specifying the search info using the keyboard?
The Spotlight search box is the one in the top, right of the menu bar - command+space. The Find search box is the one I think you're talking about - within the a Finder window.
To get focus back to the search field, hit control+tab or command+f.
If you go to System Prefs > Keyboard there's a radio button near the bottom. Select All controls and this will let you tab through the other search fields.
That said, in my testing, there's some flaws. Some of the fields aren't selectable. And if you try to select one of those, then hit command+F, you can no longer tab through the other fields anymore. You have to open a new Finder window and start over.