Can the Klein-Gordon (and maybe the Dirac) equation be derived from the path integral quantization of a given classical (supposedly relativistic) Lagrangian of particles? If so, which Lagrangian?

Usually, the Klein-Gordon equation is derived by taking the energy dispersion equation of special relativity, and promoting the energy and the momentum to their respective differential operators, in the same way the Schrödinger equation is derived from the energy dispersion equation of non-relativistic classical mechanics.

But as said, there is a second derivation of the Schrödinger equation, coming from the path integral quantization. I have not seen the same for the Klein-Gordon equation. Even worse for the Dirac equation, which is not derived from any energy dispersion equation (granted, it satisfies the energy dispersion equation when applied twice).