Yes, a red light re-exposes the cyan layer on the base side, and a blue light re-exposes the yellow layer on the emulsion side. The magenta layer can then be chemically re-exposed since the other two layers have already been developed. K-12 used a separate magenta reversal bath and wash, while in K-14 they incorporated the reversal agent into the developer itself, eliminating two steps.
Those re-exposure lights use very sharp cutoff filters to expose only the relevant layer, and adjusting the intensity of the lamps is an important factor in process control.
I'm afraid that Kodachrome may not be with us much longer. There are only a handful of labs left that run K-14 (mostly Qualex labs), and there is just not enough demand to keep justifying production on a long term basis. I think what has really been the biggest factor is the demise of super 8 movie film. When everyone stopped shooting movies and went to video tape, labs saw the super 8 volume decrease to a trickle. When I worked at the lab, movie film far surpassed slide production, and without the movies, there was not enough volume to keep the line going.