  We consider the stochastic dynamics near zero-temperature of the random
ferromagnetic Ising model on a Cayley tree of branching ratio $K$. We apply the
Boundary Real Space Renormalization procedure introduced in our previous work
(C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)) in order to derive the
renormalization rule for dynamical barriers. We obtain that the probability
distribution $P_n(B)$ of dynamical barrier for a subtree of $n$ generations
converges for large $n$ towards some traveling-wave $P_n(B) \simeq P^*(B-nv) $,
i.e. the width of the probability distribution remains finite around an
average-value that grows linearly with the number $n$ of generations. We
present numerical results for the branching ratios K=2 and K=3. We also compute
the weak-disorder expansion of the velocity $v$ for K=2.
