We introduce a broad class of families of branching random walks on a set $X$. The processes in each family are parametrized by a positive parameter $λ$ and they are monotonically increasing in $λ$ with respect to the germ order, a notion that extends classical stochastic domination. We define a general notion of critical parameter $λ(A)$ associated with a subset $A \subseteq X$ and investigate how modifications of reproduction laws affect these critical parameters.