How do you fight (out of context) and illogical debates?

I really dislike this tactic but people use it a lot. Now its perfectly reasonable to want to argue your point. If you see something wrong and you are simply arguing your side is not whats annoying. Doing it for the sake of starting conflict is another thing.

However some people like to pull this trick in a debate and just to harass people in general. They will purposely misread what you are saying and take it out of context.

Now sometimes it obvious things are not being understood correctly by both sides. However people who do it purposely make it obvious that is their intention and you can always see it. They also generally seem to come off very sarcastic.