  We consider a population with non-overlapping generations, whose size goes to
infinity. It is described by a discrete genealogy which may be time
non-homogeneous and we pay special attention to branching trees in varying
environments. A Markov chain models the dynamic of the trait of each individual
along this genealogy and may also be time non-homogeneous. Such models are
motivated by transmission processes in the cell division,
reproduction-dispersion dynamics or sampling problems in evolution. We want to
determine the evolution of the distribution of the traits among the population,
namely the asymptotic behavior of the proportion of individuals with a given
trait. We prove some quenched laws of large numbers which rely on the
ergodicity of an auxiliary process, in the same vein as \cite{guy,delmar}.
Applications to time inhomogeneous Markov chains lead us to derive a backward
(with respect to the environment) law of large numbers and a law of large
numbers on the whole population until generation $n$. A central limit is also
established in the transient case.
