The aim of transforming the Black-Scholes PDE is of course to find a form where an relatively easy solution exists. Most of the steps seem to be straightforward - please use this reference: https://planetmath.org/AnalyticSolutionOfBlackScholesPDE

...all but one, actually the last one where a convection-diffusion equation is being transformed into the basic diffusion equation.

[In the article you find it here: "Under the new coordinate system (z), we have the relations amongst vector fields ... leading to the following transformation of equation..."]

The u_x-term vanishes by some magic coordinate transformation. When you look at the derivatives they even seems wrong to me because they state that tau=s and then derive delta/delta tau = delta/delta s + some extra term (delta/delta y * -(r-1/2 sigma^2).

I simply don't get it: first how it works and second how they find that kind of transformation.