Notice, this was more than a mere preference. This was a priority. The first kind that I think is an appropriate time to change local churches, other than the Basics changing, are Family Priorities.
Objective Christian deal breakers include things like sin being embraced by the established leadership (notice that this is not about perfection, nor is it about the leader’s charisma – more on this soon). Reminder yet again, this is not about a mere preference. 
It wasn’t about how much we liked or didn’t like something about the church. It was something we prioritized in our family that the church, great as it was, couldn’t help us with.
I am going to list the Different kinds of priorities (that have come to mind so far), but first a thought:
I think if a church is missing an emphasis in an area that become a priority for you, at least a conversation or at least serious consideration should be given to whether you should be the person creating this ministry emphasis within your church.
I recommend at least making this attempt. Not every church can do everything. Certain populations seem to be called to different churches in different seasons. Right now, about 40% of the humans present at our campus on a Sunday morning are 18 or under.
That means if you aren’t in a place to prioritize ministry with children, you probably shouldn’t attend our church. Our church is 20 minutes away from the colleges in our town and you would have to drive past 25 churches almost no matter what route you took.
Our college ministry is unlikely to ever be our power center. Good news, though; a good friend of mine pastors a church across the street from one of the colleges. Anytime someone tells me their heart and passion are ministry to college students, I send them to his church – even if they are members of our church when they tell me.
Given that, it is a fair question to ask whether a certain emphasis is important for your church or whether it is another of the good things that a church could do, but not every individual church should.
Remember that the most important priorities are what we already called “Basics”. Those should probably be deal-breakers. Assuming all of those are in place, then how would we ever justify moving?
Ok, priorities:
Family priorities. This was my example, but I think Family Priorities would include all kinds of things about family decisions. Does a church support adoption and you are adopting? Do you have kids of a certain age and they seem to need special help? I am not sure what all would count here, but this seems like an important part of the conversation. Is there something our family needs to make a priority and it could require us to find a church that has that priority as well. I could see marriage issues, special needs issues, racial issues, divorce issues and others playing into this. I wonder what other examples readers can come up with?
I already gave my own example of this, but I am aware of families with mixed race adoptions, for example, who change churches so that the adopted child does not seem like the odd person out at church.
Ministry priorities. This might have fallen under Family as well, I suppose (or maybe the other way around), but they still seem distinct enough to separate them. If there is a ministry calling or passion that you have about how God has created you to minister and your church no longer emphasizes this, or another church is, or the calling is new.
Essentially, a ministry priority is taking an issue NOT with how well we are being served in this church, but how well we are ABLE to serve within and through this church.
Here are a couple of examples:
A family came to tell me that they felt called to attend the older church that had planted ours. They said that they felt like they could make a real impact in that congregation. They loved both churches but felt like God was guiding them where a young family with ministry potency was more vitally needed! We publicly commissioned them and sent them off with our blessing!
Another family, who had been missionaries in Central America, came to me and told me that another local church had initiated a Spanish speaking congregation. That church needed leaders with church experience but who spoke Spanish fluently. This family had both and they wanted to go minister there. This family had stayed for some time even though they had some biblical interpretation differences with our congregation – not Basics,but just perspectives. That minor issue had not led them to switch, but instead a ministry priority. They were called to minister to Spanish speakers and the opportunity arose in our community. We publicly commissioned them and sent them off with our blessings!
I have heard many examples of this. A passion for college students or a calling for discipleship training, a major heart for education, a well developed skill (think language 
or dedicated population)… etc.
I wrestled with other possible areas of priority – personal priorities, even theological priorities, but each of them fell either into either Family priorities, Ministry priorities or were Basics… or weren’t really good reasons to leave a church!