Speaking of debating techniques, I always let these people present whatever evidence they possess first and then I simply respond with the corresponding answers, and fill out with relevant details if necessary. Keeping it short and simple is the key really, because these people often lack attention skills and will mostly feel comfortable with very superficial explanations. Deep complex arguments are confusing and scary. But if you let them lead the conversation, you're not only showing good intentions by being a listener (which is still very important) but the topic will be defined by the believer to begin with and you'll both avoid confusion. It also becomes quite obvious to both people in the discussion when inconsistency arise, it does so because you know what the believer thinks, and the believer knows what he or she have said, so there will be no turning back once a bad argument have been exposed.
But it's interesting what you, Mick and Ross points out that instead of accepting being wrong, there might be attempts by the believer to convince him or herself that he or she has been tricked somehow instead of simply accepting being wrong, which suggests overconfidence in personal reasoning skills.
Think of it as the Dunning-Kruger effect. People who in reality lacks intelligence will think of themselves as better than they actually are. And to gain intelligence and intellect you need to be a sound sceptic, not just sceptical to external information, but apply self-criticism as a way of improving and refining your own reasoning and knowledge. The problem with conspiracy ideas is they affect very deep emotional aspects of the believer, and the fight against alleged conspiracy becomes a huge ego boost for them. Because, to be honest, from my personal observation believers in unrealistic conspiracy theories are almost always uneducated misfits, who lack social status and competence. So even if you expose their bad reasoning quite clearly for them, they might be incapable of accepting being wrong because it would mean a too large blow to the ego. So what you find instead is various psychological defence mechanisms that jumps into place.