The detector now reports variation in energy versus time for all wavelengths simultaneously. A laser beam is superimposed to provide a reference for the instrument operation. Now, you have to do the Fourier transform for every point in the interferogram. Fortunately, even a slow computer can efficiently perform this operation. All modern FT instruments are computer-interfaced.
Resolution is limited by the design of the interferometer.
The longer the path of the moving mirror, the higher the resolution. Even the least expensive FT instrument provides better resolution that all but the best CW instruments were capable of. The spectrum below is such a background scan.