We consider a multihop network where a source node must reliably deliver a set of data packets to a given destination node. To do so, the source applies a fountain code and floods the encoded packets through the network, until they reach their destination or are lost in the process. We model the probability that the destination can recover the original transmissions from the received coded packets as a function of the network topology and of the code redundancy, and show that our analytical results predict the outcome of simulations very well. These results are employed to design distributed forwarding policies that achieve a good tradeoff between the success probability and the total number of transmissions required to advance a packet toward the destination. We finally develop in detail the case where intermediate relays can inject additional redundancy in the network, provided that they have successfully decoded the source packets.