  The reproduction speed of a continuous-time branching random walk is
proportional to a positive parameter $\lambda$. There is a threshold for
$\lambda$, which is called $\lambda_w$, that separates almost sure global
extinction from global survival. Analogously, there exists another threshold
$\lambda_s$ below which any site is visited almost surely a finite number of
times (i.e.~local extinction) while above it there is a positive probability of
visiting every site infinitely many times. The local critical parameter
$\lambda_s$ is completely understood and can be computed as a function of the
reproduction rates. On the other hand, only for some classes of branching
random walks it is known that the global critical parameter $\lambda_w$ is the
inverse of a certain function of the reproduction rates, which we denote by
$K_w$. We provide here new sufficient conditions which guarantee that the
global critical parameter equals $1/K_w$. This result extends previously known
results for branching random walks on multigraphs and general branching random
walks. We show that these sufficient conditions are satisfied by periodic
tree-like branching random walks. We also discuss the critical parameter and
the critical behaviour of continuous-time branching processes in varying
environment. So far, only examples where $\lambda_w=1/K_w$ were known; here we
provide an example where $\lambda_w>1/K_w$.
