Which implementation conceptual frameworks are recommended to study organizational improvement design?

Following Popper, I am rather skeptical about absolute value of Frameworks for the advancement of science. That is, is corroboration of a 'proven' phenomenon a real contribution to science?  I think we should at least question it.   Thus the idealization of "Multiple frameworks" could also be questionable, perhaps even more. Again, following Popper, science advances by refutations (falsification) and new ways of conjecturing.

Said that, one important thing to the first part of your question: That an educational phenomenon has multiple facets, and it is entangled in a mesh of complex issues, and we should avoid lineal understanding of it is quite plausible. However, it is less convincing that a research on it requires 'multiple frameworks' all at once.  I think we can still use one single research framework. A simple example would be the Complexity Theory, which by itself allows multiple facets to entertain. (Although, as I said earlier, I am reluctant to trade me in to the idea that there is a 'super-dogmatic framework' elsewhere, and Complexity theory itself could be one such).  That a school has health problems does not mean that your research should top up a medical science framework.

Regarding your second part of the Q, the effect size  of principals (what we call vaguely as 'leaders-and-their-leadership") and teachers are rather weak according to meta-analysis (please check John Hattie's works). They are never beyond 0.4. Surprisingly, parental influence have bigger effect size (Ref. ibid.).

Hattie, J. (2009). Visible learning : a synthesis of over 800 meta-analyses relating to achievement. London ; New York: Routledge.