The only trouble is, it's not accurate.
When you're downloading something, you're taking it off a server and putting it onto your personal box. So the only one doing any downloading of brownware in this scenario is actually the toilet.
So, let's try "upload brownware". You're the host, here, but there's something counter-intuitive about saying "upload" in a largely gravity-driven operation. I don't think either download or upload works for this euphemism.
Then I hit upon another facet of it: usually that which is downloaded or uploaded is thought to be valuable in some way by the initiating party, or someone has made a request of them in some format. Is "brownware" valuable, outside of composting purposes? So I thought of what one does when one has unwanted software. Usually, one deletes it. But deletion is too swift and clean-sounding a process for the actual bodily elimination process.
Then I had it: uninstall. Depending on the complexity of the program being removed, and the quality of the un-installer (if any), un-installation of a program can be a quick event leaving no trace in the system, a slow and painful process, or quick but leaving library files and registries and the like a splattered mess with unwanted remnants about, or even with critical items removed. Yes, un-installation is an apt metaphor.
While I will probably never actually say, "Hold on, I need to go uninstall some brownware", the phrase now amuses me as intended.