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Dynamic decode-and-forward (DDF) is a version of decode-and-forward relaying in which the duration of the listening phase at relays is not fixed. In this paper, we investigate half-duplex DDF relaying based on rateless codes. The use of rateless codes allows relays to autonomously switch from listening to the source node to transmitting to the destination node. We first revisit different signal combining strategies applied at the destination node, namely energy and information combining known from literature, and propose a new combining method which we refer to as mixed combining. The different combining methods give rise to different achievable rates, i.e., constrained channel capacities, for which we provide analytical expressions. The capacity analysis reveals the conditions under which mixed combining is superior and how it can be optimized. We then consider Raptor codes as a specific implementation of rateless codes and develop a density-evolution approximation to predict the data-rate performance of these codes in DDF relaying. Furthermore, we devise an optimization of the output symbol degree distribution of Raptor codes that is mainly used to benchmark the performance of Raptor codes with a fixed degree distribution. Numerical results for exemplary three-node and four-node relay networks show that the proposed mixed combining provides significant gains in achievable data rate and that Raptor codes with a fixed degree distribution are able to realize these gains and to approach closely the constrained-capacity limits.