Inverse of transducer compositions
This is done in Chapter 6 "Decomposition of sequential functions" of 

[1] S. Eilenberg, Automata, languages and machines, Vol. B, Academic Press, New York, 1976

and there is indeed a connection with the Krohn-Rhodes decomposition. However the decomposition is weaker than just $h = f \circ g$, just like in K-R decomposition, which works up to division. See also

[2] H. Straubing, Families of recognizable sets corresponding to certain
varieties of nite monoids, J. Pure Appl. Algebra, 15, (1979) 305{318.

for application to regular languages (or Google "wreath product principle" for more recent papers).