I know wikipedia isn't the best place to get the most accurate info, but...
When stored in darkness, Kodachrome's long-term stability under suitable conditions is superior to other types of color film. Images on Kodachrome slides over fifty years old retain accurate color and density. It has been calculated that the yellow dye, the least stable, would suffer a 20% loss of dye in 185 years.[23] This is because developed Kodachrome does not retain unused color couplers. However, Kodachrome's color stability under bright light, for example during projection, is inferior to substantive slide films. Kodachrome's fade time under projection is about one hour, compared to Fujichrome's two and a half hours.[24]
I couldn't get to the Wilhelm Institute paper that's in the references, though.

And I was initially more worried about the possibility of the slide catching on fire or melting while being projected - which is probably from somewhere in my quite active imagination anyway.

Yeah, Roger's suggestion a few posts up is likely it.