In a nutshell, Capstaff's Kodachrome worked by taking a silver-gelatin negative and exposing it to a tanning bleach, not unlike a carbro bleach or a dichromate bleach. This bleach reacts with the silver and hardens the gelatin in proximity, "tanning" the highlights (a.k.a. making the gelatin much harder and less soluble; cross-linking). Areas with less silver aren't tanned as much, i.e. the shadows.
This kind of image is known as a planographic matrix; an image in gelatin defined by differences in degree of tanning. This is what you create in bromoil, by the way.
Next step, dip these in the dye baths and voila; dye absorbs more freely into the untanned gelatin & less freely into the highly tanned areas.
What R Paul is describing is in fact a different process that relies on turning the silver into something like potassium iodide(?), which is a dye mordant and then dying the films with basic dyes (F.E. Ives pioneered this I believe, for his Kromskop process). Although dichromates are something of a mordant, they're washed out in 2-color Kodachrome and thus the staining mechanism is actually an interaction between dye & gelatin.
Capstaff's process requires acid dyes; which is actually a good thing. Basic dyes are incredibly vivid but relatively fugitive in terms of light stability. Acid dyes are generally less saturated but have very good long term stability; one reason why dye-transfer prints have such excellent dark keeping. The two dyes react with gelatin totally differently, another reason why we need acid dyes for this process. Acid dyes are also used in textile dying, meaning a great variety are commercially available.
Now, the tale of these dyes is quite interesting. It all goes back to an early dye-imbibition process called Pinatype which used planographic matrices exposed under positives to create matrices capable of absorbing and transfering dye to a final support. We can call these dyes "Pinatype dyes", and what (supposedly) makes them unique is that they preferentially stain untanned gelatin over tanned gelatin. J.S. Friedman wrote quite a bit about these dyes; indicating that knowing their names or class would be beneficial to everyone. In one journal there is actually quite an impassioned discourse on this topic (which I can find if anyone is interested).
So, historically these dyes are considered "secret", or at least were so in the late 40's.
But, all hope might not be lost. It's reasonable to assume that many dyes, in fact maybe all acid dyes (or reactive dyes, or direct dyes) have this property. A good place to begin is looking at dyes suitable for dye-transfer.
I should also note that there are some other ways to go about making the gelatin "matrix". One would be to use dichromated gelatin exposed to UV under a positive (not a negative; it makes sense if you think about it and the result is the same as a negative exposed to a tanning bleach). This would also create a planographic matrix. Lastly, and perhaps the easiest, would be to expose dichromated-gelatin under a negative and etch it with hot water (like in the carbon process), creating a relief matrix. A relief matrix is exactly what a dye-transfer matrix is, and the whole thing would consist of equally tanned gelatin that would dye up in proportion to its thickness (as opposed to its degree of tanning).
So in other words, we can completely eliminate this concern of "Pinatype dyes" (if such a thing really exists) and start working with relief matrices, of which an innumerable quantity of dyes will work with. Or we can investigate if it's possible to get these same dyes to act in a Pinatype or Capstaff fashion with planographic matrices.
I hope this is helpful, if not somewhat overwhelming. These Capstaff Kodachromes are really beautiful and worth pursuing, and I plan to start working on them in the fall.