Private schools can expel (or refuse to admit) underperforming or problematic students. But in most states, charter schools have to follow the same admissions and disciplinary policies as any other public school. That includes taking special education kids. They don't get to pick and choose.
Given the large difference in cost between average students and expensive students (ESL, special needs, behavioral issues, ADHD) I expect that charter schools will find creative ways to ensure they minimize the latter. (Similar to how health insurance companies compete for young, healthy participants.)
My charter school held a lottery to see who gets admitted. This way they avoid the whole elitism via only accepting through test scores. And its standardized results were near the top of the state despite the lottery system. Now it's a different matter of what are the demographics of the applicants.
Applications, some form of interview and or testing, and rejections. Same as a university system - let students compete for spots and let schools compete for students.
I admit that there would have to be a system that forces schools to take "undesirables" who can't get accepted anywhere, which is the hard part.
Also, it brings competition to teaching... which drives people to work hard(er).
To be clear, I'm in favor of school choice. Just some thoughts.