Different pathogens spreading in the same host population often generate complex co-circulation dynamics because of the many possible interactions between the pathogens and the host immune system, the host life cycle, and the space structure of the population. Here we focus on the competition between two acute infections and we address the role of host mobility and cross-immunity in shaping possible dominance/co-dominance regimes. Host mobility is modelled as a network of traveling flows connecting nodes of a metapopulation, and the two-pathogen dynamics is simulated with a stochastic mechanistic approach. Results depict a complex scenario where, according to the relation among the epidemiological parameters of the two pathogens, mobility can either be non-influential for the competition dynamics or play a critical role in selecting the dominant pathogen. The characterisation of the parameter space can be explained in terms of the trade-off between pathogen's spreading velocity and its ability to diffuse in a sparse environment. Variations in the cross-immunity level induce a transition between presence and absence of competition. The present study disentangles the role of the relevant biological and ecological factors in the competition dynamics, and provides relevant insights into the spatial ecology of infectious diseases.