Here is my take on colour formation and how they come into play in cross processing. I am not a chemical engineer, so some of the ideas are not going to be technically spot on.
Modern colour films are technically known as integral tripack films (I know that there are more than three layers in the film- I am just talking a basic idea here) have three light sensitive layers in them. They have what are called colour formers in each layer. (Kodachrome does not - the colour formers are introduced during processing with the K-14 process.)
Light strikes the film, to effectively cause three different silver latent images, one in each colour sensitive layer, to form when the image forming exposure is made in the camera. The film is then developed. The developer is a PPD- salt type developer. The salts are complex organic compounds.
When the silver in each layer is 'developed' the developer is oxidised, proportional to the amount of light that struck that particular area. The 'salt' part of the compound is released proportional to the amount of PPD developing agent oxidised. Guess what - that salt, when it is unbound, bumps into the colour formers laying around. The colour former and oxidised salt combine to make a visible dye.
The colour formers in each layer are set up to give the three colours (this is a simplification of a very complex issue), all while react with the same salt compound in a particular developer.
Cross processing exploits that E-6 (slides) use a colour developer that is short handedly referred to as CD-3. C-41 (negatives) use a developer referred to as CD-4. There has been a CD-1 (now obsolete), and I think that there is a CD-2 used in movie film.
When you develop E-6 with CD-4 the dyes formed are not a 'designed as normal' response, but you get a distorted colur palette as an end result that many find artistically agreeable.
After the colour developer and stop bath steps the un-coupled dye formers, and exposed silver is 'bleached' away, and then any unexposed silver left in the film is fixed away.
For c-41 some processes combine the bleach and fix into a common 'blix', whille others use a separate bleach and fix.
Better E6 processes almost always use separate bleach and fix baths. For positive transparencies, there are up front steps of first developer, a black and white developer, and then a chemical reversal to chemically expose all of the sliver that was not developed by the first developer.
C-41 film has an orange mask in the film base, to deal with less than perfect dye solutions that occur when trying to make a true un-masked reversal with the available dye couplers and colour developer reactions. Early colour negative films did not have such a mask.
Ron - fee free to hop in and set me straight if any of these generalities are likely to lead others astray.