You may want to research emotional intelligence. There's lots of methods you can try but they are not quick to explain. Some examples.
Ask to explain, clarify. Someone made a derogatory comment - call it out. E.g. What do you mean by X? Are you implying that Y?
Dont get angry. Provide clear incentive to act. Increase the cost of failure. Can be as simple as CC relevant person in the email. State clearly what are the expected outcomes. Highlight the risk. Make sure others know what's going on.
Question real reason for the criticism. Is it really something in your control? Ask what they would do in such scenario. Dig into details. Point out gaps
Above all stay calm. The more the other person gets angry and confrontational while you stay calm and professional the better you look in eyes of everyone else. It makes it obvious to others who is the big baby and who can the handle pressure.
This comment ^ is the best advice. In all conflicts, it is of the utmost importance that you stay calm.
However, keep in mind that this will often infuriate or enrage an aggressor. Often an aggressor is endeavoring to provoke you so that you lash out first. When you lash out first, you lose.
I have always had a huge difficulty controlling my emotions in these situations, and so I always tried to avoid conflict. Now, I am married, and so conflict is unavoidable. It is imperative that I control how I react and respond.
I still struggle with this. I actually wish that I had some coaching on how to manage my anger, but I have had to deliberately get better with practice, and it has been hard, and sometimes even embarrassing.
Edit: One thing that I have found that helps sometimes is explaining to an aggressor why you are feeling angry immediately as you feel yourself getting angry. If something is said that hurts or offends, say so. Say, "You said that I do X, or that I am Y -- that feels unfair, inaccurate, and maybe even as though you are intentionally trying to hurt me. I wish you would find a different way to say that."
If the conflict is with a person operating in good faith, this often leads directly to an apology. If not, then you might be dealing with someone who is in fact intentionally trying to provoke you. There is no need to disclose your feelings further to such a person, but it can still be helpful to understand the nature of the attacks from which you are defending yourself.
A+ advice. I was not taught great conflict resolution skills as a kid and have some neurospiciness that makes regulation challenging at times, but my meds help and I think I’m better equipped emotionally these days than I tend to think of myself as or give myself credit for. Thank you for the advice, this is all A+ insight.