In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{ξ_{S_k}\leq s\}}-s\big) \end{equation*} where $(ξ_x, x\in\mathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{n\in\mathbb N}$ is a random walk evolving in $\mathbb{Z}^d$, independent of the $ξ$'s. In Wendler (2016), the case where $(S_n)_{n\in\mathbb N}$ is a recurrent random walk in $\mathbb{Z}$ such that $(n^{-\frac 1α}S_n)_{n\geq 1}$ converges in distribution to a stable distribution of index $α$, with $α\in(1,2]$, has been investigated. Here, we consider the cases where $(S_n)_{n\in\mathbb N}$ is either: a) a transient random walk in $\mathbb{Z}^d$, b) a recurrent random walk in $\mathbb{Z}^d$ such that $(n^{-\frac 1d}S_n)_{n\geq 1}$ converges in distribution to a stable distribution of index $d\in\{1,2\}$.