There could be multiple sub-categories for this, for specific sages that could be ran, or just all sages in one run. The ends upon receiving the Sage with the obtaining message, for specific sages, and it ends when receiving the last sage with the obtaining message for the "all sages" sub-category.

Isn't just all All Dungeons run just without the final battle?

Basically, but the main focus her would be for specific sages