When you run out of things to do in one round, you should automatically move on to the next. I was stuck multiple times only because I didn't realize that going to the next round was entirely manual so I kept trying to find something to do and that really kills the pacing for the game.
Either that, or maybe put some kind of glow or icon over characters that you can interact with to continue the plot. Without it, it's just "click everyone until you can't do anything else" over and over again, and that's not fun.