  Branching processes form an important family of stochastic processes that
have been successfully applied in many fields. In this paper, we focus our
attention on controlled multi-type branching processes (CMBPs). A Feller-type
diffusion approximation is derived for some critical CMBPs. Namely, we consider
a sequence of appropriately scaled random step functions formed from a critical
CMBP with control distributions having expectations that satisfy a kind of
linearity assumption. It is proved that such a sequence converges weakly toward
a squared Bessel process supported by a ray determined by an eigenvector of a
matrix related to the offspring mean matrix and the control distributions of
the branching process in question. As applications, among others, we derive
Feller-type diffusion approximations of critical, primitive multi-type
branching processes with immigration and some two-sex branching processes. We
also describe the asymptotic behaviour of the relative frequencies of distinct
types of individuals for critical CMBPs.
