  We study branching Markov chains on a countable state space (space of types)
$\mathscr{X}$, with the focus on the qualitative aspects of the limit behaviour
of the evolving empirical population distributions. No conditions are imposed
on the multitype offspring distributions at the points of $\mathscr{X}$ other
than to have the same average and to satisfy a uniform $L \log L$ moment
condition. We show that the arising population martingale is uniformly
integrable. Convergence of population averages of the branching chain is then
put in connection with stationary spaces of the associated ordinary Markov
chain on $\mathscr{X}$ (assumed to be irreducible and transient). This is
applied, in particular, to the boundaries of appropriate compactifications of
$\mathscr{X}$. Final considerations consider the general interplay between the
measure theoretic boundaries of the branching chain and the associated ordinary
chain.
