Context. The magnetic field of galaxies is believed to be produced by internal dynamo action, but can be affected by motion of the galaxy through the surrounding medium. Observations of polarized radio emission of galaxies located in galaxy clusters have revealed noticeable features of large-scale magnetic configurations, including displacements of the magnetic structures from the optical images and tails, which are possible imprints of ram pressure effects arising from motion of the galaxies through the intracluster medium. Aims. We present a quantitative dynamo model, which attempts to describe the above effects. In contrast, to the traditional problem of a wind affecting a body with a prescribed magnetic field, we investigate how a non-magnetized wind flow affects a magnetic field that is being self-excited by galactic dynamo action. Methods. To isolate the leading physical effects, we exploit a simple dynamo model that can describe relevant effects. In particular, we use what is known as the "no-z" approximation for the mean-field dynamo equations. Results. In a suitable parametric range we obtain displacements of the large-scale magnetic field, as well as magnetic tails. However, the specific details of their locations are quite counterintuitive. The direction of displacement is perpendicular to, rather than parallel to, the wind direction. The point at which the tail emerges from the galaxy depends on details of the model. The tail is eventually directed downstream. In the simplest case the magnetic tail begins in the region where the wind decreases the total gas velocity. Any wind that penetrates the galaxy modifies the intrinsic dynamo action. These features are different from those found in ram-pressure models. Conclusions. Any determination of galactic motion through the cluster medium from observational data needs to take the effects of dynamo action into account.