A few interesting articles in Business Insider have outlined some of the research into how to win an argument, and it's simpler than you think.

When people sense your points don't match their world view they shut down the logical functions and switch to combative mode. Whichever way you try to steer them over to your point of view won't work as they're preparing to battle you on any point. If, however, you present them their own viewpoints to ridiculous conclusions they are much more willing to reconsider.
They gave the example of a study by Tel Aviv University with a group of Israelis in 2013, just prior to the Israeli elections. One of their fundamental beliefs is that they are the most moral society so saying that "Israel should continue the conflict so its citizens could continue to feel moral" you are taking their fundamental beliefs to absurd conclusions. 30% were willing to reconsider their views and the effects lasted for a year as they were also more likely to vote for more moderate candidates in the following elections.
Instead of inviting people to ask why they believe a certain thing and allowing them to outline all the reasons immigration should be stopped etc ask how they would achieve what their arguing for. Ask them to take you through all the steps of implementing their beliefs from start to finish, step by step, as to how their policy will achieve what they believe it will.
A Colorado Psychologist, Philip Fernbach, in his paper entitled
"Political Extremism is Supported by an Illusion of Understanding" invited people to debate about complex issues such as healthcare and Iran. They either outlined all the reasons they were right or the had to explain how they would implement their viewpoints from start to finish. The first group maintained their extreme views but the second group had softer views and also rated their understanding of the subjects as lower.
So in the end, people won't logically consider your side unless it's presented in the context of their world view, and inviting people to outline the practicalities of the policy views and their conclusions will get them to actually think through their opinions on a topic and realise they don't understand it as much as they thought they did.