For the last few years, the research interest in magnetoelectric (ME) effect, which is the cross-coupling between ferroelectric and magnetic ordering in multiferroic materials, has experienced a significant revival. The extensive recent studies are not only conducted towards the design of sensors, actuators, transducers, and memory devices by taking advantage of the cross-control of polarization (or magnetization) by magnetic (or electric) fields, but also aim to create a clearer picture in understanding the sources of ME responses and the novel effects associated with them. Here we derive analytical models allowing to understand the striking and novel dynamics of ME effects in multiferroics and further confirm it with atomistic simulations. Specifically, the role of strain is revealed to lead to the existence of electroacoustic magnons, a new quasiparticle that mixes acoustic and optical phonons with magnons, which results in resonances and thus a dramatic enhancement of magnetoelectric responses. Moreover, a unique aspect of the dynamical quadratic ME response under a magnetic field with varying frequencies, which is the second harmonic generation (SHG), has not been discussed prior to the present work. These SHGs put emphasis on the fact that nonlinearities should be considered while dealing with such systems.