The current answer to the last question isn't magnets but .. lasers.
Free Electron lasers beam power to the climber, which converts the
energy into mechanical energy (wheels or treads). IIRC, a FEL has
been designed that can do the job.

Please give me a link to this design. If you are talking about simply a
beamer that could perhaps be used in an unknown design that incorporates
aiming (you know, to hit that elevator), suitable transfer medium, and
receiver system, which can operate on these lurdicous lenghts, then dont
bother.