Well, it seems that nothing is more stable than a carbon print because it is actually pigment trapped inside the gelatin. You can think of it like making a photograph with painter's pigments, and these substances can be incredibly stable. Vermeer's paintings haven't faded for 350 years!
Dyes on the other hand aren't as stable, and will fade overtime. Dye-transfer prints do have excellent dark storage properties, but their light-fastness just isn't as great. Acid dyes, which are used in dye-transfer, are quite stable compared to, say, basic dyes, which are quite fugitive in short amounts of time (yet more brilliant!). So it depends on the specific dye being utilized, and tradeoffs are made between hue, light-fastness and their imbibition properties.
I'm no expert, but I think you could hang an UltraStable print outside, facing the sun and it wouldn't appreciably fade in your lifetime. (this could be a bit of an exaggeration... I'm taking poetic license. . . but maybe not)