  By a random process with immigration at random times we mean a shot noise
process with a random response function (response process) in which shots occur
at arbitrary random times. The so defined random processes generalize random
processes with immigration at the epochs of a renewal process which were
introduced in [Iksanov et al. (2017). Bernoulli, 23, 1233--1278] and bear a
strong resemblance to a random characteristic in general branching processes
and the counting process in a fixed generation of a branching random walk
generated by a general point process. We provide sufficient conditions which
ensure weak convergence of finite-dimensional distributions of these processes
to certain Gaussian processes. Our main result is specialised to several
particular instances of random times and response processes.
