Ignition and combustion of an initially spherical pocket of fuel in motion relative to a hot oxidizing atmosphere is studied. The model considers finite rate chemistry represented by an irreversible one-step reaction. Attention is focused on the development of the ignition process, which, in the end, typically leads to the establishment of a diffusion flame. For moderate values of a suitably defined Damkohler number, three stages are identified in the burning process. Stage I corresponds to an induction period during which a mixing layer is formed around the fuel kernel. This stage ends up with a thermal runaway close to the back of the kernel. Stage II involves the propagation of a chemical front in the mixing layer: initiated at the point where the thermal runaway occurs, a first flame travels around the fuel kernel towards the nose, triggering during its travel a premixed radial inwardly propagating flame. The second stage ends when ignition reaches the nose. Stage III corresponds to an established diffusive burning, which is most active at the front surface. As the Damkohler number, Da, is increased, stage I and II shrink leading to a practically spherical ignition. In this limit of large Da, the ignition time becomes independent of the Reynolds number. Conversely, for sufficiently low Da, stage III (i.e., diffusive burning) is absent. In this case, the fuel is practically consumed by reaction at the back of the kernel, after premixing at the front mixing layer with the oxidizer. The flame responsible for the burning occupies a quasi-steady stable position close to the maximum cross section position. Furthermore, the results provide a good appreciation of the dynamics of the combustion process. For example, it is found for moderate Da that a significant acceleration-deceleration of the fuel pocket takes place during ignition due to the pressure increase caused by gas expansion. Finally, with the aid of an order-of-magnitude analysis, a synthesis of most of the physical results described above is achieved by delimiting different domains in the Da-Re plane.