Dynamic properties of collinear phases realised in magnets with easy-plane single-ion anisotropy in an external magnetic field perpendicular to an 'easy plane' are studied. A phase diagram characterised by the existence of a cascade of reorientational phase transitions induced by the field and the single-ion anisotropy is constructed. The spectrum of collective excitations, their damping, scattering amplitudes, critical fields at low temperatures, position of multicritical points, and critical behaviour peculiarities at T=0 in the vicinity of the second-order phase transition lines and of the multicritical points are investigated.