The .22 Win Auto is shorter (.915") than the .22 LR (.975"), so the carrier may need some work, but it is the other way. Another factor would be that the rim diameter of the Win Auto is bigger (.310") than the .22 LR (.275"), so a change might affect feeding as well as extraction/ejection, but I don't think the problem would be unsolvable.
I would see no point in exchanging one obsolete cartridge for another, as .22 WRF is just about as scarce as .22 Win Auto, plus it is longer than either the .22 WRF or the .22 LR, so I think conversion problems would be more complicated. The .22 WRF rim size is .295", between the .22 Win Auto and the .22 LR, so its use might still require bolt face work.
This is the kind of job that I think could be done and work out, but if I were to try it I would want to do a lot of study before going to work.
BTW, the .22 Win Auto fired a 45 grain bullet at 1055 fps, so I think there would be no problem with the .22 LR 40 grain at 1150 or so. If I were to do the job, I would recommed standard velocity .22 LR, though.
Jim