My understanding of how /dev/urandom works is that the kernel collects true randomness from a variety of sources, and mixes them together to remove statistical bias. It does this at regular intervals to refresh the entropy pool.
The CSPRNG then takes a subset of these bits to generate random bits for consumption.
My question: in the course of dipping into the entropy pool, does the CSPRNG also update the entropy pool or is that done only by the OS?
The CSPRNG is updated from the input pool from time to time (at most every 5 minutes by default). It does not feed back into the input pool since it is derived entirely from it in the first place.
Prior to 4.8, the input pool would periodically feed into the blocking pool and non-blocking pool at long enough intervals to provide a "catastrophic reseed" that mixes in enough entropy that it could not be brute forced even given knowledge of the prior state. These two pools were identical, with the exception that the blocking pool would track entropy estimates and block if the estimate drops too low. Since 4.8, the non-blocking pool has been replaced by a CSPRNG based on the ChaCha20 stream cipher.
From Documentation and Analysis of the Linux Random Number Generator, version 2.5
