Branching random walk with trapping zones

We consider a branching random walk with values in a certain set $\mathbb{S}$, where the branching mechanism is different according to whether particles (individuals) are in a certain so called trapping set $A?\mathbb{S}$ or not. We are then interested, under different scenarios, in properties of either the transient random measure describing distribution of individuals on $\mathbb{S}$ over time or its asymptotic behaviour.