So, hopped in, had fun, bought the game. I started a new campaign, and we kicked butt. The AI dudes even surrendered... and the battle kept going. Even with no enemies left and apparently 100% control of the map, the round won't end and move the campaign forward. Is there something a player needs to do to end the match, or what?

Any help is very welcome.

There should be points around the map that lets you progress the campaign. In the map, look for Doors and hover over them - they should tell you what map they'll lead to. You can even see the maps you can go to in the top-right of the map (a map of the continent, so to speak). Just go near the door and voila, it'll teleport you to the next map.

Awesome, I had complete missed the door icon. Thanks!