We study the steady-state and transient responses of a second-order elastic plate by implementing an asymptotic analysis of the three-dimensional equations with respect to two geometric characteristics seen as parameters: the thickness of the plate and an inner material length. Depending on their ratio, four different models arise. Conditions under which Reissner-Mindlin kinematics may appear are discussed while the influence of crystalline symmetries is studied. The transient situation is solved through Trotter's theory of approximation of semi-groups of operators acting on variable spaces.