  Consider a sequence of i.i.d. random variables $X_n$ where each random
variable is refreshed independently according to a Poisson clock. At any fixed
time $t$ the law of the sequence is the same as for the sequence at time 0 but
at random times almost sure properties of the sequence may be violated. If
there are such \emph{exceptional times} we say that the property is
\emph{dynamically sensitive}, otherwise we call it \emph{dynamically stable}.
In this note we consider branching random walks on Cayley graphs and prove that
recurrence and transience are dynamically stable in the sub-and supercritical
regime. While the critical case is left open in general we prove dynamical
stability for a specific class of Cayley graphs. Our proof combines techniques
from the theory of ranching random walks with those of dynamical percolation.
