The white is an optical trick, and the trick runs on ultraviolet
Cotton canvas and mesh are not white. They are a slightly cream colour, and the difference is made up by optical brighteners — fluorescent compounds that absorb ultraviolet light in the 340 to 400 nanometre range and re-emit it as blue light around 430 to 450 nanometres. That blue is added to the light the fabric reflects, and the eye reads the mixture as white. It is a genuine trick, and it is why a new trainer looks brighter than white paint: it is emitting light in a region where a painted white surface has nothing to give.
Which tells you precisely why sunlight is the enemy. Ultraviolet both excites the brightener and, over months, degrades it. Once the brightener is gone there is no blue being added, so the underlying cream shows through — and that is what "my white shoes went yellow" usually means. The fabric has not changed colour. The trick has stopped working.
And the second half of the problem is the wash. Washing powder is alkaline, and a rinse cycle does not remove all of it; what stays behind dries inside the fibre. On a sunny windowsill the water evaporates, the residue concentrates, and ultraviolet accelerates its oxidation, so you get yellow patches with crisp edges — the edges are where the liquid stopped running. So "wash in powder and put them in the sun" does two harmful things at once: it leaves the oxidising residue in the fabric, and it destroys the brightener that was covering for it.
Three yellows, three answers
Crisp patches that appear as they dry are residue and they are reversible. Wash again with a small amount of detergent, then rinse three times rather than once — this is the whole repair — wrap the shoe in paper and dry it in shade. The paper absorbs the yellow that migrates out and blocks a layer of ultraviolet; the shade does the rest.
An even cream over the whole upper, laces yellow too, is the brightener gone. Oxygen bleach — sodium percarbonate at about 20 g per litre of warm water, an hour — will lift the oxidised material and brighten it one step. It will not bring back the fluorescence, so do not expect the shoe you bought. From then on: paper and shade, never the windowsill.
A brown rubber sole and midsole is a different chemistry entirely, and this is the one place where the sun helps. Rubber and TPU yellow through oxidation of the polymer and its antiozonants, and the yellowing can be partly reversed by hydrogen peroxide plus ultraviolet — the same reaction sold as retrobright for old computer cases. Coat the rubber in peroxide paste, wrap it in cling film so it does not dry out, and leave it in direct sun for four to six hours. It will come back in a few months, because it is ageing rather than dirt, but the difference is dramatic and it costs almost nothing.