Our national discourse on school segregation, whether income- or race-/ethnicity-based, tends to focus on the separation of students between schools within districts. There are good reasons for this, including the fact that the majority of desegregation efforts have been within-district efforts. Sometimes lost in this focus, however, is the importance of segregation between districts.
This distinction can be confusing, so consider a large metro area with a central city district surrounded by a group of suburban districts. There may be extensive racial/ethnic segregation of students between schools within those districts, with students of color concentrated in some schools and their White peers concentrated in others. But total segregation across the entire metro area is also a function of segregation between districts - i.e., the degree to which students of certain races or ethnicities are concentrated in some districts and not others (e.g., students of color in the city, white students in the suburbs). In a sense, if we view diversity as a resource, there are multiple "chokepoints" at which that resource is distributed down to the next level—from states to metro areas to districts to schools—and this can exacerbate segregation.
A recent working paper provides one of relatively few pieces of recent evidence suggesting that, in addition to racial and ethnic segregation between districts and between schools within districts, there may be an additional important "layer": segregation within schools.