  In Li (2011), Example 2.2, the notion of a multi-type continuous-state
branching process (MCSBP) was introduced with a finite number of types, with
the countably infinite case being proposed in Kyprianou and Palau (2017). One
may consider such processes as a super-Markov chain on a countable state-space
of types, which undertakes both local and non-local branching. In Kyprianou and
Palau (2017) it was shown that, for MCSBPs, under mild conditions, there exists
a lead eigenvalue which characterises the spectral radius of the linear
semigroup associated to the process. Moreover, in a qualitative sense, the sign
of this eigenvalue distinguishes between the cases where there is local
extinction and exponential growth. In this paper, we continue in this vein and
show that, when the number of types is finite, the lead eigenvalue gives the
precise almost sure rate of growth of each type. This result matches perfectly
classical analogues for multi-type Galton--Watson processes.
