This is an extremely interesting question. I'm going to take a different approach to the question by focusing on both personality traits and leadership theories (e.g. authentic leadership, transformational leadership, servant leadership etc) to answer whether those two distinct areas can influence leaders' children's development. I will admit that I didn't come across specific studies on the actual family situations of leaders and the developmental impacts of the leader's personality traits towards their children.
Experiential learning and development by children
See Murphy and Johnson (2011) about how early life experiences and parenting styles can shape the development of a children in relation to general development and leadership potential.
but parents undoubtedly work to inﬂuence their children's thoughts on
leadership, through role modeling leadership and through the leader
behaviors and experiences they encourage in their children ... “[Parents…] contribute to their children's religious beliefs,
intellectual and occupational interests, feelings of self-esteem or
inadequacy, adherence to traditional or modern notions of masculinity
and femininity, helpfulness to others, skills, and values (Beer,
Arnold, & Loehlin, 1998; Kruger, Hicks, & McGue, 2001; McCrae et al.,
2000)” (Wade & Tavris, 2008, p. 502).
Servant leadership promotes spirituality and servant behaviours in leaders, i.e. serve your employees first before you serve the company. This type of behaviour from a leader would most likely influence the parent's value and style of parenting. For example, leaders who are recognised as servant leaders may be generous and selfless - this may spill over into their parenting style to be compassionate and caring towards their children which in turn, influences their children's development into similar characters.
ADDED: Destructive leaders, narcissism and potential impact on children
Behaviours of destructive leaders tend to be correlated to narcissism (Padilla et al., 2007). Assuming that these behaviours manifest in the home/family environment, there is some evidence that child narcissism can arise from the parent's control over them (Horton, 2010):
There is also substantial evidence supporting the object relations
viewpoint that child narcissism comes from a parent’s selﬁsh use of
the child that is manifest in excessive or inconsistent parental
control. For example, both Horton and colleagues (2006) and Miller and
Campbell (2008) found that psychological control correlated positively
with maladaptive narcissism...
Similarly, destructive leadership involves an element of the urge to control (Padilla et al., 2007):
... destructive leadership involves control and coercion rather than
persuasion and commitment (Howell & Avolio, 1992; Sankowsky, 1995)
I would dare say that if we compared destructive leaders (core trait: narcissism) and servant leaders (core: selflessness) - both would exhibit different parenting styles that both impact a child's development. Children of the former group may grow up to develop childhood narcissism which may stretch out to adulthood. Children of the latter group may grow to develop selfless behaviours in childhood and adulthood. Yet, I wish there would be a study demonstrating this.
- Murphy, S.E. & Johnson, S.K. (2011). The beneﬁts of a long-lens approach to leader development: Understanding
the seeds of leadership. Leadership Quarterly, 22, 459-470
- Horton, R. S. (2010). On environmental sources of child narcissism: Are parents really to blame? (pp. 125-143), In Barry, C, Kerig, P. K, Stellwagen, K. K, & Barry, T. T (Eds.) Narcissism and Machiavellianism in Youth: Implications for the Development of Adaptive and Maladaptive Behavior. Washington, D.C.: APA Press.
- Padilla, A., Hogan, R. & Kaiser, R.B. (2007). The toxic triangle: Destructive leaders, susceptible followers,
and conducive environments. The Leadership Quarterly, 18, 176-194