We develop a criterion for transience for a general model of branching Markov chains. In the case of multi-dimensional branching random walk in random environment (BRWRE) this criterion becomes explicit. In particular, we show that Condition L of Comets and Popov  is necessary and sufficient for transience as conjectured. Furthermore, the criterion applies to two important classes of branching random walks and implies that the critical branching random walk is transient resp. dies out locally.
Sebastian Müller. "A criterion for transience of multidimensional branching random walk in random environment." Electron. J. Probab. 13 1189 - 1202, 2008. https://doi.org/10.1214/EJP.v13-517