The branching random walk on a regular graph turns out to be particularly easy to analyse using results for the corresponding simple random walk. In this way, one can show that there is an intermediate phase of weak survival if and only if the graph is nonamenable. No such simple analysis holds more generally, and it is known that the nonamenability equivalence does not extend to general connected graphs of bounded degree (although we observe that it does hold for such graphs if the branching random walk is modified in a certain natural way). The most important general class of (bounded degree, connected) graphs for which it is thought that the equivalence may hold is that of quasi-transitive graphs: we show that this is indeed the case.