Done right, cross-examination helps your debate presentation three ways. First, it helps clarify your opponent’s extremist position. Second, it gives you opportunity to discredit his evidence (or lack thereof) and expose his flawed logic. Third, it allows you to gain information you can use later in the debate.
The rules for cross-x are simple: The person doing the questioning controls the exchange at that point in the debate. For example, when it’s your turn to cross-x your opponent, you may interrupt him at any point and move to the next question. While you shouldn’t be rude (give him reasonable time to answer, no more), you also shouldn’t let him ramble on.
Stephanie Gray of the Canadian Centre for Bioethical Reform presents a textbook example of how it's done: