We present an exact treatment of wave propagation in some inhomogeneous thin films with highly space-dependent dielectric constant. It is based on a space transformation which replaces the physical space by the optical path. In the new space, the dispersion equation is that of a normal progressive wave. We will show that the dispersion properties of such films are plasma- or waveguide-like, the characteristic frequency being determined by the spatial characteristics of the dielectric constant's variations only. The theory is scalable, so that it can be applied in any wavelength range: optical, IR, radiofrequency, etc. depending only on the characteristic space scales. Several applications will be presented, concerning the reflection properties of such films (broadband anti-reflection, or dichroic coatings) or to the propagation and transmission through the film. We will show that depending on the type of space dependence, an incident wave can either propagate or tunnel through such films. We will investigate the behaviour of the light group-velocity and tunneling time inside or through such films. Though we can reproduce the phase-time saturation corresponding to the Hartman effect, analysis of the group velocity in the tunneling case shows no sign of superluminal propagation. A strong frequency dependence can be obtained in some situations, which allows to anticipate a strong reshaping of brodband laser pulses.