Suppose that X = (X t , t ≥ 0) is either a superprocess or a branching Markov process on a general space E, with non-local branching mechanism and probabilities P δx , when issued from a unit mass at x ∈ E. For a general setting in which the first moment semigroup of X displays a Perron-Frobenius type behaviour, we show that, for k ≥ 2 and any positive bounded measurable function f on E, lim t→∞ g(t)E δx [ f, X t k ] = C k (x, f) where the constant C k (x, f) can be identified in terms of the principal right eigenfunction and left eigen-measure and g(t) is an appropriate determinisitic normalisation, which can be identified explicitly as either polynomial in t or exponential in t, depending on whether X is a critical, supercritical or subcritical process.