We analyze a notable class of states relevant to an immiscible bosonic binary mixture loaded in a rotating boxlike circular trap, i.e., states where vortices in one species host the atoms of the other species, which thus play the role of massive cores. The resulting effective Lagrangian resembles that of charged particles in a static electromagnetic field, where the canonical momentum includes an electromagnetic term. The simplest example is a single-vortex system. While a massless vortex can only precess uniformly, the presence of a sufficiently large filled vortex core renders such precession unstable. A small core mass can also enhance the orbit radius or lead to small radial oscillations, which are, in turn, clear evidence of the associated inertial effect.