Yes, to mumble to. If you want to clarify the direction of his mumbled remarks, you'll have to use to.
He mumbled to the wind.
If you don't need the direction or recipient of his underbreath comments, then it would be He mumbled about something.
The bounces expression is not familiar to me. From context, it seems he has persuaded you to go to the party, in a playful way. If it was an online discussion (a language all its own), he might have "bounced" or "bumped" an invitation to the top of a forum by making a comment on a buried and forgotten thread.
Student or Learner