Is it right to try and get someone to change their mind

with respect to their conscience or should you respect them for not doing so?
I put the question up on Twitter yesterday - and then asked people not to respond because it is much too difficult a question to debate on Twitter. It would lead to all sorts of misunderstandings.
I came across a problem yesterday. I was asked by someone I know quite well,
"Cat, how do you get someone to change their mind?"
It is not the sort of question I like to be asked. There is no definitive answer to that sort of question - apart from "it is very difficult".  There is another question I need to ask as well, "Should I be trying to change somebody's mind?" 
Rather than try and answer the question I asked what the problem was.
"Well, you know D....? He's opposed to gay marriage and it is causing all sorts of problems in the office."
I do know D. His attitude was causing problems in the office? 
"Oh, it's not him. It's other people. They keep trying to get him to change his mind."
I suspect the person I was talking to had also tried and she was looking for some sort of reassurance that they were "doing the right thing".
I am afraid I said, "It's none of their business."
And it isn't. D doesn't try to change their minds.  When asked, I have heard him say, "I believe marriage is between a man and a woman. I am also happy for same sex couples to have legally recognised civil unions."
I feel if someone makes that sort of statement then it should be the end of the story. It most certainly isn't my business to try and change his or her beliefs. 
When I was stopped yesterday there were several stories in the media about similar issues. There is a heated discussion going on over whether one of the local councils (local authority or municipality) is right in flying the rainbow flag. 
A married man with three children has resigned his position as the manager of another council because they also decided to fly the rainbow flag. He says he doesn't oppose them doing it but does not wish to be seen to personally support the decision because he too believes marriage is between a man and a woman. 
And, of course, the issue of gay marriage has again been brought up in our federal parliament. It is proving divisive there as well. The media is making a major issue of it, exploiting differences of opinion. A former federal politician told me recently that he was glad he didn't have to be there when the issue is eventually voted on. 
It shouldn't be a divisive issue. I also believe it is wrong to try and get someone to change their mind over the issue. D  is not homophobic. He believes in equal opportunities for all - something I have seen him put into place in his workplace. He's actually very much more tolerant than many other people I know. He tries to live according to his Christian convictions - the religion he would never dream of ramming down anyone's throats. 
I don't thing he is wrong or right or anything else. He has never told me what I should believe and I am not going to tell him.
It's not my business and it is not the business of those who are trying to get him to change his beliefs. 
So, why do they think it is their business? What are they trying to do?