"What are the French kindergarten teachers flying on?"
Translation:Min repülnek a francia óvónők?
min is mi "what" + -n for the case meaning "on..."; thus min "on what".
mit is mi "what" + -t for the accusative case; thus "what" as a direct object (e.g. Mit csinálsz? "What are you doing?" - the "what" is the object of "doing"; you are asking about the thing that is being done).
mi is "what" in the nominative case; you could use it for asking about the subject of a sentence, for example: Mi van a házban? "What is in the house?"
The verb has to appear directly after the question word, since the focus of the sentence is on that word. So "Min repülnek a francia óvónők?" and "A francia óvónők min repülnek?" are possible solutions.