Let $(Z_n,n\geq 0)$ be a supercritical Galton-Watson process whose offspring distribution $μ$ has mean $λ>1$ and is such that $\int x(\log(x))_+ dμ(x)<+\infty$. According to the famous Kesten \& Stigum theorem, $(Z_n/λ^n)$ converges almost surely, as $n\to+\infty$. The limiting random variable has mean~1, and its distribution is characterised as the solution of a fixed point equation. \par In this paper, we consider a family of Galton-Watson processes $(Z_n(λ), n\geq 0)$ defined for~$λ$ ranging in an interval $I\subset (1, \infty)$, and where we interpret $λ$ as the time (when $n$ is the generation). The number of children of an individual at time~$λ$ is given by $X(λ)$, where $(X(λ))_{λ\in I}$ is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that $\mathbb E(X(λ))=λ>1$ for all $λ\in I$. This allows us to define $Z_n(λ)$ the number of elements in the $n$th generation at time $λ$. Set $W_n(λ)= Z_n(λ)/λ^n$ for all $n\geq 0$ and $λ\in I$. We prove that, under some moment conditions on the process~$X$, the sequence of processes $(W_n(λ), λ\in I)_{n\geq 0}$ converges in probability as~$n$ tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.