We devise and evaluate five methods for fountain-coded broadcast distribution of a file from a source to multiple destinations in an ad hoc wireless network that consists of half-duplex packet radios. The methods differ in their use of intermediate nodes, their use of forwarding, and their reliance on a network spanning tree. All five methods employ continued fountain coding to prevent nodes from receiving duplicate fountain-coded packets. We derive an analytical approximation for the throughput of fountain-coded broadcast file distribution in a four-node network with time-varying radio links modeled by independent two-state Markov chains, and we show that our approach to fountain-coded file distribution gives throughput that is very close to the approximation. We employ simulations to examine larger networks in which each radio link has correlated Rayleigh fading and the radios use adaptive modulation and channel coding.