thermodynamics - Are all diffusion-like processes described as wave-like in relativity-compatible formulations?

Citing from Wikipedia's article on relativistic heat conduction:

For most of the last century, it was recognized that Fourier equation (and its more general Fick's law of diffusion) is in contradiction with the theory of relativity, for at least one reason: it admits infinite speed of propagation of heat signals within the continuum field. [...] To overcome this contradiction, workers such as Cattaneo, Vernotte, Chester, and others proposed that Fourier equation should be upgraded from the parabolic to a hyperbolic form,

$$\frac{1}{C^2}\frac{\partial^2 \theta}{\partial t^2} +\frac{1}{\alpha}\frac{\partial \theta}{\partial t}=\nabla^2\theta$$ also known as the Telegrapher's equation. Interestingly, the form of this equation traces its origins to Maxwell’s equations of electrodynamics; hence, the wave nature of heat is implied.

It appears to me that the PDEs describing any other diffusion process –for instance, the Fokker–Planck equation for Brownian motion– will also assume an infinite speed of propagation. Then, if my intuition is correct, they'll be incompatible with SR, and will have to be "upgraded" to hyperbolic, wave-like equations.

If this were a general rule, would we have, for instance, a relativistic wave equation for Brownian motion? It appears unlikely... Is there, then, any example of diffusion-like/dispersive equation whose form "survives" into a relativity-compatible description?

Edit:

I'll add a broader reformulation of the question, as suggested by a @CuriousOne comment:

Can we find a first order equation that models the finite velocity limits or are we automatically being thrown back to second order equations? Is there a general mathematical theorem at play here about the solutions of first vs. second order equations?