  Let $X_1, X_2,\ldots$ be random elements of the Skorokhod space
$D(\mathbb{R})$ and $\xi_1, \xi_2, \ldots$ positive random variables such that
the pairs $(X_1,\xi_1), (X_2,\xi_2),\ldots$ are independent and identically
distributed. The random process $Y(t):=\sum_{k \geq
0}X_{k+1}(t-\xi_1-\ldots-\xi_k)1_{\{\xi_1+\ldots+\xi_k\leq t\}}$,
$t\in\mathbb{R}$, is called random process with immigration at the epochs of a
renewal process. Assuming that the distribution of $\xi_1$ 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+).
