  Let $(X_1, \xi_1), (X_2,\xi_2),\ldots$ be i.i.d.~copies of a pair $(X,\xi)$
where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$
and $\xi$ is a positive random variable. Define $S_k := \xi_1+\ldots+\xi_k$, $k
\in \mathbb{N}_0$ and $Y(t) := \sum_{k\geq 0} X_{k+1}(t-S_k) 1_{\{S_k \leq
t\}}$, $t\geq 0$. We call the process $(Y(t))_{t \geq 0}$ random process with
immigration at the epochs of a renewal process. We investigate weak convergence
of the finite-dimensional distributions of $(Y(ut))_{u>0}$ as $t\to\infty$.
Under the assumptions that the covariance function of $X$ is regularly varying
in $(0,\infty)\times (0,\infty)$ in a uniform way, the class of limiting
processes is rather rich and includes Gaussian processes with explicitly given
covariance functions, fractionally integrated stable L\'evy motions and their
sums when the law of $\xi$ belongs to the domain of attraction of a stable law
with finite mean, and conditionally Gaussian processes with explicitly given
(conditional) covariance functions, fractionally integrated inverse stable
subordinators and their sums when the law of $\xi$ belongs to the domain of
attraction of a stable law with infinite mean.
