  We consider a class of multitype Galton-Watson branching processes with a
countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a
block lower Hessenberg form. For these processes, the probability
$\boldsymbol{q}(A)$ of extinction in subsets of types $A\subseteq
\mathcal{X}_d$ may differ from the global extinction probability
$\boldsymbol{q}$ and the partial extinction probability
$\tilde{\boldsymbol{q}}$. After deriving partial and global extinction
criteria, we develop conditions for
$\boldsymbol{q}<\boldsymbol{q}(A)<\tilde{\boldsymbol{q}}$. We then present an
iterative method to compute the vector $\boldsymbol{q}(A)$ for any set $A$.
Finally, we investigate the location of the vectors $\boldsymbol{q}(A)$ in the
set of fixed points of the progeny generating vector.
