This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ² < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law B _{p v n} ^{*N _n bef} which depends on the total population size N _n ^bef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ², as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.