My doubt is the following: What does the occurrence of an exponential expansion of the early universe has to do with the Cosmic microwave background (CMB) homogeneity since those photons (that compose the CMB) correspond to a much latter epoch and were emitted from regions that at that time were already causally disconnected?
The argument is that
(mixing happens) => (inflation happens) => (mixed regions are out of causal contact, but have no way to change their local temperature)
In this scenario, it doesn't matter when the photons are emitted. Their apparent homogenity is an effect of the left most step.