  As an integral part of our culture and way of life, language is intricately
related to migrations of people. To understand whether and how migration shapes
language formation processes we examine the dynamics of the naming game with
migrating agents. (i) When all agents may migrate, the dynamics generates an
effective surface tension, which drives the coarsening. Such a behaviour is
very robust and appears for a wide range of densities of agents and their
migration rates. (ii) However, when only multilingual agents are allowed to
migrate, monolingual islands are typically formed. In such a case, when the
migration rate is sufficiently large, the majority of agents acquire a common
language, which spontaneously emerges with no indication of the surface-tension
driven coarsening. A relatively slow coarsening that takes place in a dense
static population is very fragile, and most likely, an arbitrarily small
migration rate can divert the system toward quick formation of monolingual
islands. Our work shows that migration influences language formation processes
but additional details like density, or mobility of agents are needed to
specify more precisely this influence.
