The Court was not blind to the fact that desegregation/integration would not and could not be done with the snap of a finger. There was not only the logistics of desegregation, but full knowledge of the numerous legal challenges that were already pending, but also would surely be filed. Keep in mind there was more than one Brown v. Board case; they're referred to as Brown I (1954), Brown II (1955), and Brown III (1992). Brown I is the famous one. Short, sweet, and to the point, it lays out the broad constitutional principle we're all familiar with today. However, Brown II dealt with the realities of desegregation and Brown III was about a concern of de facto segregation happening despite the prohibition.

Here is an excerpt from Brown II that outlines the Court's understanding that desegregation would be a process:

Full implementation of these constitutional principles may require solution of varied local school problems. School authorities have the primary responsibility for elucidating, assessing, and solving these problems; courts will have to consider whether the action of school authorities constitutes good faith implementation of the governing constitutional principles. Because of their proximity to local conditions and the possible need for further hearings, the courts which originally heard these cases can best perform this judicial appraisal. Accordingly, we believe it appropriate to remand the cases to those courts.

Later in the opinion, the Court directed lower courts to administer desegregation "with all deliberate speed" which is easier said than done.

The Court basically acknowledged that school administrators would have a hell of a time desegregating, and left it in the hands of the local District Courts (and appropriate state courts) to oversee desegregation considering the sheer amount of challenges or requests for injunctive relief filed by both school administrators and opponents. Indeed, the Court issued numerous opinions in the subsequent decades dealing with the implementation of Brown I.