It's a year and a half since "alazio" kindly provided the solution above, and this is still clear as mud in the app nor have I see this solution in the documentation.
I just lost a half hour trying to get a batch to work. I had selected the 1st option, "Save to original location," plus "authorized" that location after having selected it. "OK" remained greyed out. After finding this solution, I used the 2nd option, "Save into," selected the path, and it's finally working. (Or not. I'm converting .dng's to .jpg, but the .jpg's are saving very underexposed, yet they individually export fine, so now I've another puzzle to solve. Exporting via batch is not an option.)
So, why does the 1st option exist? And why did I just wasted a half hour plus put time into this forum because Affinity, a year and a half later, hasn't fixed or at least clarified this in the app, let alone the documentation?