Your second example is a transition from a Conflict (the fight) to a Contest (the chase).

> Transitioning to a Contest or Challenge — [Fate Core, page 173](https://fate-srd.com/fate-core/conflicts#transitioning-to-a-contest-or-challenge)
> ----
> You may find yourself in a conflict scene where the participants are no
longer interested in or willing to harm one another, because of some change
in the circumstances. If that happens, and there’s still more to resolve, you
can transition straight into a contest or challenge as you need. In that case,
hold off on awarding the end-of-conflict fate points and whatnot until
you’ve also resolved the contest or challenge.

Your first example is a Conflict potentially transitioning to another kind of Conflict, if the other party wishes to continue.  The question is, what is the other party's response if you start throwing punches?  They could stop arguing and [Concede the Conflict (page 167)](https://fate-srd.com/fate-core/conflicts#conceding-the-conflict). They could flee, which is a transition to a Contest, as above. If they fight back, you're continuing the Conflict but now with different attack skills and consequences, per the definition of a [Conflict (page 154)](https://fate-srd.com/fate-core/conflicts).

>  **As long as
the characters involved have both the intent and the ability to harm one
another, then you’re in a conflict scene.**  Conflicts are either physical or mental in nature, based on the kind of
harm you’re at risk of suffering.