We discuss the validity and accuracy of thin-disc asymptotic solutions of the mean-field dynamo equations, using a new numerical code to perform calculations for embedded disc models, with high spatial resolution on a nonuniform grid. We show that there is reasonable agreement between conventional linear asymptotic solutions (using vacuum boundary conditions) and numerical results, provided that the concept of effective disc thickness is introduced. This result is of more general significance in the context of disc dynamo models, in that it is necessary to redefine the dynamo number of models with vacuum boundary conditions in order to apply them to realistic discs without sharp boundaries. We also consider the effects of competition between eigenmodes with different growth rates localized in different regions of the disc. A front associated with the faster growing eigenmode propagates through the disc and eventually dominates the solution. Under typical galactic conditions, the frontal velocity is typically a few km s−1. The propagation of fronts outwards from central regions can provide seed fields for subsequent mean field dynamo action that are stronger than conventionally assumed.