If by "drama proper" you mean modern drama, then yes, Doctor Faustus is a transitional piece. On the surface, it resembles medieval morality plays. There is a definite line between good and evil. There is a strong religious element, as Faustus is tempted by Mephistopheles and the forces of Satan while still given chances to redeem himself by asking God for forgiveness. However, certain elements complicate the morality play factors.
Unlike medieval morality plays, there is a greater emphasis on individual action and psychology in Doctor Faustus. The forces of evil are strong, but the human will is even stronger. At any point, Faustus could have returned to God and regained his salvation, but he never does. Such occurs not because the villains stop him, but because Faustus's character psychology is so rooted in pride that he is never willing to admit he has done wrong. In a standard morality play, Faustus would have been a flatter, less individual character and more of a bland everyman.
By making Faustus an individual, the play becomes a tragedy, not merely a series of moralistic tableaux. It is this complexity of character which makes Marlowe's play closer to modern drama.