  The classical Galton--Watson process works with a fixed probability of
fission at each time step. One of the generalizations is that the probabilities
depend on time. We consider one of the most complex and interesting cases when
we do not know the exact probabilities of fission at each time step - these
probabilities are random variables themselves. The limit distributions of the
number of descendants are described in terms of generalized integral and
differential functional equations of the Schr\"oder type. There are no more
analogs of periodic Karlin-McGregor functions, which were very helpful in the
analysis of the asymptotic behavior of limit distributions for the classical
case. We propose some approximate asymptotic methods. Even simple cases of
random families with one or two members lead to nice asymptotics involving,
interesting problems related to special functions and special constants. One of
them, Example 2 is already announced on
\href{https://math.stackexchange.com/questions/4748129/asymptotics-of-sequence-of-rational-numbers}{this}
and
\href{https://mathoverflow.net/questions/458885/simple-integral-equation}{this}
sites. Finally, the phenomenon of why the oscillations in the main asymptotic
term usual for the classical case become rare in the case of random
environments is discussed.
