Frequently when working with church boards or wider ministry leadership groups there will be one gentleman sitting in the back of the room with his arms crossed, a hostile glare and no matter how much laughter there is in the room, never cracking a smile. He is saying loudly and non verbally, "You cannot convince me to change my mind." He is usually correct and I don't try.
What he illustrates is that there are people in every church or organization who will never be convinced to go where you desire to go. And, since leadership requires us to initiate change to respond to the environment and opportunities around us, you will always have people who don't want to respond if it means change. In fact, there is no conversation with them: their mind has been made up and that is that!
We have various names for people like this. On the change scale, they are called laggards. The scale moves from innovators who initiate change to early adapters who respond to change quickly to middle adapters who take some time to process the change to late adapters who take even longer and finally laggards who are change adverse. Another name for laggards is "squeaky wheels" because not only are they change adverse but they can be loud and obnoxious about it (ever met one in a congregational meeting?).
This is not a judge of character, a mistake we often make. People are wired to respond to change differently and that in itself is neither right or wrong although attitudes like the one noted above are not always very gracious or helpful.
What do you do with the unconvincible? Love them but don't try to convince them. Rather you appeal to the people who will respond to change if given a valid reason and a value that is higher then their resistance to change. The unconvincible are unconvincible. It is the convincible who are your audience. Appeal to them and they will go with you if you make a reasonable case for the proposed change.
In any change process, remember that the loud voices do not necessarily reflect the majority of the group. Often they represent a small minority of laggards but their voices can be intimidating. Don't be intimidated. Run a good process of dialogue and communication and if you will likely gain the majority of folks. In the process don't marginalize the laggards but recognize that their resistance to change usually comes from how they are wired, not bad character. In fact, if you love them, they will often be your most loyal supporters even when resistant to change. Listen to them, love them and stay connected to them relationally and they will usually love you back.