The dynamics of freely standing thermotropic smectic-A films are studied in the isothermal, incompressible limit via a continuous hydrodynamic description. The role of permeation in the films, the structure of the hydrodynamic normal modes, and the form of the autocorrelation functions for the smectic layer and order-parameter fluctuations are discussed. We find two characteristic lengths l(d)=sqrt[alphad/B] and l(c)=sqrt[eta(2)(3)d/8rhoalpha] associated with the dynamic behavior of the system, where alpha is the surface tension, d is the film thickness, B is the elastic constant for layer compression, eta(3) is the layer sliding viscosity, and rho is the density of the liquid crystal. The crossover from filmlike to bulklike behavior is controlled by l(d) alone; the crossover from overdamped to underdamped dynamics when the in-plane length scale is large compared to l(d) is controlled by l(c) alone.