Here are two abstracts from the References section in the link I provided. These, plus the other papers referenced, probably suffice to establish the basis upon which I inferred, although I'm open to suggestions for other, maybe better, ways to generalize it:

"This paper presents the results of a 20-year longitudinal study of moral judgment development. The study represents an attempt to document the basic assumptions of Kohlberg's cognitive-developmental theory of moral judgment. Subjects were 58 boys aged 10, 13, and 16 at time 1 and were approximately equally divided at each age by social class and sociometric status. Sociometric and socioeconomic groups were equalized for intelligence. The study included six testing times-the original interview and five follow-up interviews administered at 3-4-year intervals. At each testing time subjects were individually interviewed on their judgments about nine hypothetical moral dilemmas. Interviews were stage scored according to Forms A, B, and C of the Standard Issue Scoring Manual. All scoring was done blind by individual dilemma. Data are presented on test-retest, alternate form, and interrater reliability for Standard Issue Scoring. Validity of the instrument is discussed. It was found that subjects proceeded through the developmental stages in the hypothesized sequence. No subject skipped a stage in the sequence and only 4% (6) of the adjacent testing times showed downward stage change. This percentage was less than downward change on test-retest data. Moral judgment interviews also showed a high degree of internal consistency in stage scores assigned with the great majority of the interviews receiving all their scores at two adjacent stages. Factor analyses by dilemma and moral issue showed a single general moral stage factor. Moral judgment was found to be positively correlated with age, socio-economic status, IQ, and education. Stage scores in childhood were significantly correlated with adulthood scores. The results of this study were interpreted as being consistent with a cognitive-developmental stage model. Subjects seemed to use a coherent structural orientation in thinking about a variety of moral dilemmas. Their thinking developed in a regular sequence of stages, neither skipping a stage nor reverting to use of a prior stage. The Standard Issue Scoring System was found to be reliable, and it was concluded that it provides a valid measure of Kohlberg's moral judgment stages."

-- Colby, Anne; Gibbs, J.; Lieberman, M.; Kohlberg, L. (1983). A Longitudinal Study of Moral Judgment: A Monograph for the Society of Research in Child Development. Chicago, IL: The University of Chicago Press. ISBN 978-99932-7-870-2.

"Several issues concerning Gilligan's model of moral orientations and Kohlberg's models of moral stages and moral orientations were examined in a longitudinal study with 233 subjects (from 78 families) who ranged in age from 5 to 63 years. They participated in 2 identical interviews separated by a 2-year interval. In each interview, they discussed hypothetical dilemmas and a personally generated real-life dilemma, which were scored for both moral stage and moral orientation (both Gilligan's and Kohlberg's typologies). Results revealed few violations of the stage sequence over the longitudinal interval, supporting Kohlberg's moral stage model. Sex differences were almost completely absent for both Gilligan's and Kohlberg's moral orientations, although there were clear developmental trends. Hypothetical and real-life dilemmas elicited different moral orientations, especially in terms of Kohlberg's typology. The interrelations between the 2 models of moral orientations were generally weak, indicating that they are not synonymous."

-- Walker, Lawrence, J. (February 1989). A longitudinal study of moral reasoning. Child Development. 60 (1): 157–166. doi:10.2307/1131081. JSTOR 1131081. PMID 2702866.

Kohlberg's developmental stages theory is one of the most tested, subjected to falsification attempts, and well corroborated psychological theories of the last 60 years. Knowing it, and how the stages are distributed (among adults it follows a normal distribution), provides a solid auxiliary reference point to, e.g., reasonings based on IQ distribution, and what it implies for rationality efforts and related subject areas.

Now, I don't actually know whether Sagan, Feynman, Dawkins, or Hawkins incorporated the results of modern, well supported Neo-Piagetian studies such as Kohlberg's in their own psycho-social models and proposal. Maybe they did, and those results are already accounted for in their reasoning. But if they didn't, then I think we may think of those proposals as lacking key information and, by extension, effectiveness.