How does multi-rate processing work?
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.