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For a continuous-time Bienaym\'e-Galton-Watson process, $X$, with immigrationand culling, $0$ as an absorbing state, call $X^q$ the process that resultsfrom killing $X$ at rate $q\in (0,\infty)$, followed by stopping it onextinction or explosion. Then an explicit identification of the relevantharmonic functions of $X^q$ allows to determine the Laplace transforms (atargument $q$) of the first passage times downwards and of the explosion timefor $X$. Strictly speaking, this is accomplished only when the killing rate $q$is sufficiently large (but always when the branching mechanism is notsupercritical or if there is no culling). In particular, taking the limit$q\downarrow 0$ (whenever possible) yields the passage downwards and explosionprobabilities for $X$. A number of other consequences of these results arepresented.

Some harmonic functions for killed Markov branching processes with immigration and culling