I do not believe that we change one another's minds. Rather, we exchange information as material by which we each may change our own minds. Reason is for changing (or resisting change to) our own minds, not the minds of others. Furthermore, I don't believe that we should change our minds in the middle of a debate (short of dropping non-essential lines of argument), but should use the debate as a time to present our side as best we can and allow our opponent to do the same, examining them in comparison to one another. The debate is a collaborative examination of ideas, using contest to better display each side. It's after the debate is over that we sit down alone with what the debate has given us and contemplate how we shall or shall not change our own minds.
That said, I can provide you with an example of a friend coming to me and telling me that something I'd said to him years before had had an impact on him and he had changed his mind on the issue. Before, he hadn't believed in free will. It wasn't really a debate, I was just chatting with him about my disagreement with the argument, "If it turns out that all of your choices were actually made by a Martian, then in fact you never really had free will." I corrected the argument: "If it turns out that all of your choices were actually made by a Martian, then in fact you were always a Martian." He is now a compatibilist like myself.