(I'm not an astronomer, my understanding is not absolute, please correct if I'm wrong) I believe our Sun is a type G, yellow dwarf. A red dwarf is usually much smaller, if considering "red dwarf" being a type M, then usually no more than 0.6 solar masses, usually much smaller, and a tiiiiny fraction of the luminosity of Sol. Brown dwarfs are not massive enough to fuse hydrogen, where red dwarfs generally are. According to https://en.wikipedia.org/wiki/Red_dwarf, above 0.25 solar masses, they should evolve into red giants, otherwise will shrink into a white dwarf once hydrogen fusion is ceased. The universe isn't old enough to have any red dwarf of 0.25 solar masses to do so, however as their lifespans are so long.

> Once that's over (thinking in the larger timescale of things) and it becomes a brown dwarf,

Our sun will never become a brown dwarf. Stars have their life-cycle and nomenclature. You cannot made your own willy-nilly.