Suppose John has a foreign accent and Mark does not. The two get into an argument with each other. At some point, Mark has no good reply, so he simply mimics John in his accent to insult him. Is there a word for such behavior?
Yes, it's called mocking. Mocking is a bad substitute for a well reasoned reply.
Mocking a person publicly with hurtful words, which is really what Mark is doing with his imitation accent, is rhetorically speaking argumentum ad hominem, meaning "an argument against the man." You know, if you lack an intelligent riposte, just call your opponent stupid, or a jerk, or a nincompoop, or an a**hole.
You'd be surprised (or maybe you wouldn't) by how common the ol' ad hominem is in public discourse. A person who uses it consistently is scraping the bottom of the barrel, rhetorically speaking. He or she needs to grow up a bit and sharpen their thinking and speaking skills!
In the case of Mark and John, with his mocking, Mark is in essence insulting John by mimicking his accent, which is tantamount to calling John a derogatory name.