  It is a common method for proving weak convergence of a sequence of
time-homogeneous Markov processes towards a time-homogeneous Markov process
first to show convergence of the corresponding infinitesimal generators and
then to check some additional conditions. The aim of the present paper is to
investigate convergence properties of discrete infinitesimal generators of
appropriately scaled random step functions formed from a multi-type continuous
state and continuous time branching process with immigration. We also present a
convergence result for usual infinitesimal generators of the branching
processes in question appropriately normalized.
