Sometimes when you’ve won an argument you can help the person who lost save face by letting them get the last word in. I think this can be particularly helpful with kids.
You can help move things along by letting a kid stomp away mumbling. This seems counterintuitive and I know it’s tempting to holler, “What’s that? Who do you think is such a mean mom? Get back here this minute!” but don’t give into it. If you pull them back, “What was that you said?” you’ve entered into a power struggle. Power struggles are tar pits; do not get stuck in them.
You can pretend not to hear it even if you do, which is going to help things get back to normal more quick