We derive an asymptotic lower bound on the Bayes risk when N identical quantum systems whose state depends on a vector of unknown parameters are jointly measured in an arbitrary way and the parameters of interest estimated on the basis of the resulting data. The bound is an integrated version of a quantum Cramér-Rao bound due to Holevo (1982), and it thereby links the fixed N exact Bayesian optimality usually pursued in the physics literature with the pointwise asymptotic optimality favoured in classical mathematical statistics. By heuristic arguments the bound can be expected to be sharp. This does turn out to be the case in various important examples, where it can be used to prove asymptotic optimality of interesting and useful measurement-and-estimation schemes. On the way we obtain a new family of "dual Holevo bounds" of independent interest.