  We study an extension of the so-called defective Galton-Watson processes
obtained by allowing the offspring distribution to change over the generations.
Thus, in these processes, the individuals reproduce independently of the others
and in accordance to some possibly defective offspring distribution depending
on the generation. Moreover, the defect $1-f_n(1)$ of the offspring
distribution at generation $n$ represents the probability that the process hits
an absorbing state $\Delta$ at that generation. We focus on the asymptotic
behaviour of these processes. We establish the almost sure convergence of the
process to a random variable with values in $\mathbb{N}_0\cup\{\Delta\}$ and we
provide two characterisations of the duality extinction-absorption at $\Delta$.
We also state some results on the absorption time and the properties of the
process conditioned upon its non-absorption, some of which require us to
introduce the notion of defective branching trees in varying environment.
