Let $X_1, X_2,\ldots$ be random elements of the Skorokhod space $D(\mathbb{R})$ and $ξ_1, ξ_2, \ldots$ positive random variables such that the pairs $(X_1,ξ_1), (X_2,ξ_2),\ldots$ are independent and identically distributed. The random process $Y(t):=\sum_{k \geq 0}X_{k+1}(t-ξ_1-\ldots-ξ_k)1_{\{ξ_1+\ldots+ξ_k\leq t\}}$, $t\in\mathbb{R}$, is called random process with immigration at the epochs of a renewal process. Assuming that the distribution of $ξ_1$ is nonlattice and has finite mean while the process $X_1$ decays sufficiently fast, we prove weak convergence of $(Y(u+t))_{u\in\mathbb{R}}$ as $t\to\infty$ on $D(\mathbb{R})$ endowed with the $J_1$-topology. The present paper continues the line of research initiated in Iksanov, Marynych and Meiners (2015+).