keyboardmacos
I change the setting ‘Full Keyboard Access’ (the Keyboard settings in the ‘Keyboard Shortcuts’ tab) from ‘Text Boxes and Lists Only’ to ‘All controls’. Now, I am able to press Tab to moving focus, as illustrated below.
When I hit Enter, pop up a window, instead close the window.
My desired action: hit Enter → move to ‘Don't Save’ → Enter → Close the window.
Go to...
and select All controls under Full Keyboard Access
The Spotlight search box is the one in the top-right of the menu bar (commandspace). The Find search box is the one I think you're talking about, within a Finder window.
To get focus back to the search field, hit controltab or commandf.
Another option is found in System Prefs > Keyboard > Shortcuts. There's a radio button near the bottom labeled "Full Keyboard Access". "All Controls" will let you tab through the other search fields without modifier keys.
That said, in my testing, there're some flaws. Some of the fields aren't selectable, and if you try to select one of those and then hit commandf, you can no longer tab through the other fields anymore—you have to open a new Finder window and start over.
Best Answer
Hitting Return will activate the default command (the blue button). Pressing the space bar will activate the selected command (blue ring around it).