How does multi-rate processing work? In the case with a 1-D input like audio, how can a system with one input do many different kinds of processing at different sampling rates and still, at the end of the processing, produce one waveform? For example an ADC/DAC reading and writing at 48kHz, but has an FFT running at 8kHz or another set of processes running at 16 kHz? Is it a matter of decimating and interpolating? How would all of these processes line up?
You are on the right track.  You can use decimation and interpolation (or resampling, which is a combination of both decimation and interpolation) to change the sample rate.  For instance, if you start with 48 kHz samples you can decimate by 6 to get down to 8 kHz, do some processing, then interpolate by 2 to get to 16 kHz, do some processing, then interpolate by 3 to get back to 48 kHz.

"Lining up the processes" is not in and of itself difficult.  You can decimate and interpolate whenever you want.  Doing it in a way that makes sense and is useful, on the other hand, requires knowing more about what you start out with and what you want to accomplish.